Partial Differential Equations

Entropy production governs irreversibility and uncertainty in both physical and information-theoretic systems. While Physics-Informed Neural Networks (PINNs) successfully solve differential equations, current architectures remain inherently domain-specific. The extraction of domain-invariant entropy representations across fundamentally different physical laws remains unexplored. This paper introduces a unified Physics-Informed Deep Learning (PIDL) framework that simultaneously enforces differential equation residuals and information-theoretic bounds within a single neural architecture. We demonstrate this framework via two canonical studies: (i) a thermodynamic continuous stirred-tank reactor (CSTR) model solving governing ODEs, where a Softplus constraint strictly enforces the Second Law of Thermodynamics; and (ii) an information-theoretic financial market model solving the inverse Fokker-Planck PDE to infer latent drift and diffusion coefficients, guaranteeing diffusion positivity via a Softplus constraint while naturally inducing Shannon entropy. Three model variants are evaluated: two domain-specific baselines and one shared-encoder architecture. The PIDL framework guarantees absolute thermodynamic admissibility with zero Second-Law violations and exhibits exceptional data efficiency, retaining >90% predictive accuracy using merely 30% of available training data. Furthermore, a post-hoc Ruppeiner Riemannian geometric analysis of the learned entropy surface successfully identifies thermodynamic phase instabilities. This methodology provides a robust, domain-agnostic architecture for physics-constrained entropy modeling, advancing applications in sustainable process design and quantitative financial risk assessment.

Full waveform inversion (FWI) reconstructs heterogeneous material properties from receiver data but remains computationally demanding. Physics-informed neural networks (PINNs) and their domain-decomposed variants (FBPINNs) offer a mesh-free alternative but face convergence challenges when representing complex velocity fields. We present a hybrid quantum-classical FBPINN for acoustic FWI, bringing together quantum computing and classical machine learning, in which the decomposed wavefield network and the global velocity network are implemented as classical-to-quantum pipelines terminating in parameterized quantum circuits (PQCs). The PQCs are realized as differentiable JAX statevector simulators, enabling end-to-end automatic differentiation through the classical PINN, the quantum circuit, and the physics-informed loss. On a geophysical anomaly benchmark, the quantum hybrid reaches a lower L1 velocity error than the primary classical FBPINN baseline in approximately 8x fewer training iterations, despite using approximately 33% fewer trainable parameters, and it outperforms all 15 classical hyperparameter variants tested. A second benchmark (checkerboard) demonstrates the generality of the inversion pipeline, confirming that the quantum hybrid architecture can recover structured spatial variations beyond the localized anomaly benchmark. Our framework is broadly applicable to wave-based inverse problems beyond geophysics, including medical ultrasound tomography and non-destructive evaluation.

Mineral prospectivity modelling (MPM) underpins exploration economics, yet most operational pipelines reduce to data-driven classifiers trained on shallow surface proxies. Such models are blind to the subsurface physics that actually localises ore: heat advection, fluid flow, and lithology-dependent precipitation. We present Korzhinskii-Net, a 2-D radial physics-informed neural network (PINN) that couples Darcy flow, advective-diffusive heat transport, and a softplus-saturated reaction rate into a single differentiable forward model, weakly supervised by surface and remote-sensing proxies. The network is named after Dmitri S. Korzhinskii (1899-1985), whose theory of infiltration metasomatism provides the physical scaffold. We evaluate Korzhinskii-Net on six ore provinces spanning three commodity classes - Udokan (sandstone-hosted Cu), Sukhoi Log, Olimpiada, and Berezovskoye (orogenic Au), Vorontsovskoye (Carlin-type Au), and Dalnegorsk (skarn polymetallic) - under a fair, leakage-controlled 5-fold cross-validation protocol with hard ring-shaped negatives and baseline proxy features disabled. Korzhinskii-Net attains a mean PR-AUC of 0.708 versus 0.235 for the strongest classical baseline (support vector machine), and a mean fractional rank of 0.036 versus 0.475. The improvement is consistent across all six provinces and three commodity systems, suggesting that physics-informed differentiable simulators, even when constrained only by global open-data proxies, can recover localisation patterns that pure feature-based learners systematically miss. We release the full pipeline and evaluation harness as open source.

Neural operators provide fast surrogate models for PDE simulations, but standard architectures often treat geometry and discretization as secondary to field data. Physical states are usually represented as grid-channel stacks, even when different quantities naturally belong on vertices, edges, faces, cells, boundaries, or interfaces and must satisfy compatibility constraints. We propose Cellular Sheaf Neural Operators, a discretization-aware framework for structure-preserving neural PDE surrogates. The method represents PDE states on oriented cell complexes, couples local feature spaces through learned restriction maps, and uses incidence/Hodge-informed message passing to follow computational geometry. Learned update heads pass through coboundary or flux maps, allowing selected constraints to arise from cell-complex structure rather than only from loss penalties. For magnetohydrodynamics, this yields face-based magnetic-flux updates driven by edge electromotive fields and finite-volume-style fluid updates driven by learned face fluxes and cell sources. On turbulent MHD and fusion-equilibrium surrogate tasks, the method improves structure-sensitive diagnostics, including rollout behavior, divergence control, spectral error, and equilibrium-regression accuracy. These results indicate that cellular-sheaf structure is a useful inductive bias for neural PDE surrogates in constrained multiphysics systems.

Inference and control in engineered physical systems pay a heavy physics cost at deployment: state estimators, inverse-problem solvers, model-predictive controllers, schedulers, and observers are often not closed-form and must re-solve a numerical optimization per instance, with the operator re-supplied each time. Physics-informed learning moves this cost to training, but uses a single encoder pathway whose latent geometry de-learns under fine-tuning and admits no quantitative transfer guarantee. We propose an asymmetric two-pathway architecture that resolves both issues. A teacher encoder consumes privileged dense states from a high-fidelity simulator and represents the system through operator-polynomial features stable under spectral perturbation; a student encoder learns the same latent geometry from sparse field data and operator descriptors. At deployment the teacher is discarded, and the frozen student runs in a single forward pass with a transfer certificate. The design connects to privileged-information learning, knowledge distillation, and cross-modal distillation, but targets cross-instance transfer rather than fixed-instance prediction: topology and operator may change, while the latent task does not. We establish sufficient and near-necessary transfer conditions via Wasserstein proximity between latent laws, yielding a zero-shot error bound, and develop a finite-sample certification protocol with active expansion when coverage is incomplete. The framework applies wherever a system admits an operator with reportable spectrum. On power-system estimation, it achieves zero-shot transfer to 100 unseen topologies, a 95% certificate pass rate, accuracy competitive with topology-aware Newton--Raphson, and sub-millisecond inference. These results suggest asymmetric pathways plus operator-anchored latent geometry provide a foundation for certified zero-shot inference and control.

Physics-Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill-conditioned loss landscapes inherited from the underlying differential operator. We study PINNs augmented with a pointwise data-fidelity term, added at a few points in the domain to the standard residual and boundary losses. We show that this supervision term acts as an operator-level preconditioner: for suitable weights, our comparison bounds guarantee a substantially smaller condition number than under the standard PINN loss, independently of how the pointwise labels are obtained. For a broad class of PDEs admitting a Feynman-Kac (FK) representation, we generate such labels by Monte Carlo averages of the FK functional, resulting in what we call ``FK-PINNs", and using the excess risk decomposition approach, we derive non-asymptotic $L^2(Ω)$-error bounds for FK-PINNs with $\tanh$ activation trained by finitely many steps of gradient descent. Along the way, we establish pseudo-dimension bounds for first- and second-order derivatives of $\tanh$ neural networks, which are of independent interest and, to the best of our knowledge, new. Numerical experiments on Poisson, Schrödinger, mean exit time, and committor problems corroborate the theory, and show that FK-PINNs can successfully solve PDEs for which standard PINNs exhibit severe failure modes.

We introduce a perturbative approach for nonparametric instrumental variable (NPIV) estimation. By drawing inspiration from perturbation theory in physics, we extend standard kernel ridge methods with systematic higher perturbation order corrections that significantly improve estimation accuracy. Spectrally, the perturbation introduces mixing between different eigenmodes of the expectation integral operator, which becomes especially useful when the integral equation is ill-defined. One source for such ill-definedness can be the curse of dimensionality. Our method performs across various dimensionality regimes, particularly when the dimensionality parameter $β$ which is defined through the number of samples $n$ and dimension $d$ as $n^β= d$, becomes large. Experimental results show that our first-order perturbative corrections can reduce prediction error by up to 99% in high-dimensional ill-defined cases ($β> 0.7$) compared to standard ridge regression approaches. The performance improvement is maintained across a wide range of dimensions, with the advantage becoming more pronounced as dimensionality increases.

We present a neural-network-based framework for the solution of three-dimensional boundary value problems where the solution is expressible in terms of harmonic potentials. The approach leverages the Whittaker integral formula, which allows representing the solution through functions that are holomorphic with respect to a suitable complex variable. These functions are subsequently approximated using holomorphic neural networks, which guaranty fulfillment of the holomorphicity requirement. A key feature of the proposed formulation is that the governing partial differential equations (PDEs) are satisfied exactly by construction. Therefore, in contrast to standard physics-informed neural networks, no residual minimization of PDEs is required in the interior of the domain, and training is based exclusively on boundary collocation points. The method is validated against three-dimensional Laplace and linear elasticity problems, where, in the latter case, displacement and stress fields are expressed via the Papkovich-Neuber potentials. The numerical results show an accurate approximation of both scalar and vector fields, with errors remaining controlled throughout the domain. Overall, the work demonstrates that the incorporation of analytical structures into neural network architectures provides a natural and effective framework for the meshless approximation of three-dimensional boundary value problems while preserving the underlying properties of the governing equations.

Solving time-dependent partial differential equations (PDEs) is an important problem in computational science and engineering. Physics-informed neural networks (PINNs) learn PDE solutions from governing equations. However, accurately capturing temporal evolution remains challenging. Recent sequence-model-based approaches parameterize time evolution using general-purpose sequence models, which capture temporal dependencies but do not explicitly encode the structured dynamics of PDE solutions. In addition, their memory requirements can scale unfavorably with sequence length and resolution, limiting applicability in large-scale or high-dimensional settings. This work introduces a PINN approach that incorporates oscillatory state-space dynamics to represent the modal structure of PDE solutions. The proposed method leverages a linear-oscillator-based temporal evolution, together with a PDE-aware spectral basis in space. This design enables closed-form spatial differentiation and consistent enforcement of boundary conditions. The method is evaluated on forward, inverse, and high-dimensional PDE problems, including cases up to 100 spatial dimensions. The results show improved accuracy and reduced memory usage compared to recent sequence-model-based PINN approaches. Overall, this work highlights the benefits of incorporating structured dynamical priors into the temporal evolution of neural PDE solvers and suggests designing more physics-aligned and computationally efficient PINN architectures.

Nonlinear conservation laws are at the heart of many of the most important dynamical systems in science and engineering. In practical applications, such systems are often subject to various sources of uncertainty, e.g. due to sparse or noisy measurements. Inferring physical quantities and fields of interest then becomes an ill-posed problem which both classical numerical methods and modern deep learning-based methods struggle to treat appropriately. Recent work has framed classical numerical methods as Bayesian inference under Gaussian process priors, resulting in a physics-aware treatment of uncertainties. Following this line of work, we develop a novel numerically conservative method for uncertainty-aware simulations of nonlinear conservation laws. We use recent sparse approximation techniques to scale up to large-scale forward and inverse problems. For forward simulation, we inherit the accuracy of classical solvers while providing structured uncertainty quantification. On inverse problems, we recover posteriors over nonparametric source fields in seconds -- outperforming neural baselines that take minutes to produce a less accurate point estimate.

Riemannian diffusion models generalize score-based generative modeling to manifold-supported data via stochastic diffusion equations on the manifold. However, training requires sampling from and differentiating the manifold heat kernel, which is rarely available in closed form beyond a few highly symmetric manifolds. We propose a general approach that approximates the heat kernel by directly solving the manifold heat equation with a physics-informed neural network (PINN). Given an explicit manifold specification, we choose a coordinate system, derive the corresponding heat (Fokker--Planck) equation and a short-time asymptotic approximation, and then train a PINN to learn the log heat kernel. The resulting surrogate enables both forward noising (heat-kernel sampling) and conditional-score evaluation for denoising score matching. We demonstrate the method on diverse manifolds including $S^2$, $SO(3)$, $\mathrm{SPD}(n)$, and permutation-quotiented point clouds.

Solving high-frequency partial differential equations (PDEs) with neural networks is notoriously difficult due to the spectral bias of conventional architectures. We propose the Multi-Scale Separable Fourier Neural Network (MS-SFNN), a framework designed to overcome this limitation by explicitly encoding multi-scale Fourier features within a separable representation. The network factorizes the solution into $d$ single-coordinate subnetworks with fixed, randomly initialized weights; these subnetworks are combined via element-wise products to form a rich set of basis functions. This separable construction scales linearly with the problem dimension, thus inherently alleviating the curse of dimensionality. Crucially, each subnetwork is equipped with trainable scaling factors coupled with cosine activations, providing an adaptive mechanism for multi-scale frequency selection that endows the model with strong spectral approximation capability. The PDE solution is expressed as a linear combination of these learned basis functions, and the combination coefficients are determined by solving a large-scale least-squares system. To resolve the memory bottleneck in high-frequency or three-dimensional settings, we replace automatic differentiation (AD) with analytical derivatives of the basis functions and use a memory-efficient batched QR decomposition for solving the least-squares systems efficiently. Extensive numerical experiments demonstrate that MS-SFNN achieves superior accuracy and substantially outperforms state-of-the-art methods, including Physics-Informed Neural Network (PINN) and the Separated-Variable Spectral Neural Network (SV-SNN).

We introduce the Laplace-Fourier Neural Operator (LFNO), a unified framework for modeling dynamical systems across transient and steady-state regimes by integrating the spectral advantages of Laplace and Fourier Neural Operators. LFNO employs a dual-branch architecture that explicitly decomposes system dynamics into transient and steady-state components. We evaluate LFNO on nine benchmarks, including three ODE systems (Duffing, Lorenz, and Pendulum) and six PDE systems (Euler-Bernoulli beam, Heat, Reaction-diffusion, Brusselator, Burgers, and Navier-Stokes). LFNO significantly outperforms existing operators on ODE systems, where transient dynamics dominate, and consistently surpasses LNO while achieving performance competitive with FNO on PDE benchmarks. Furthermore, LFNO offers improved stability and physical interpretability through its component-wise decomposition. These results demonstrate that LFNO provides a robust and unified approach for learning complex dynamical systems across multiple temporal scales.

Understanding the human brain requires access to its microscopic tissue architecture. Diffusion magnetic resonance imaging (MRI) provides the only noninvasive window into whole-brain microstructure in vivo, yet reliable quantitative mapping remains confined to specialized research settings requiring dense sampling and optimized acquisition protocols. To address this gap, we present a physics-informed generative microstructure network (PIGMENT) that learns a universal generative prior of human brain microstructure and adapts it zero-shot to each participant's measured data to recover subject-specific maps. Trained on 11375 scans spanning multiple sites, vendors, and field strengths, PIGMENT enabled reliable quantitative mapping for tensor, kurtosis, and NODDI models across external datasets from five independent centers. It remains effective where conventional fitting becomes unreliable, recovering meaningful maps from extremely sparse acquisitions while supporting downstream tractography and structural connectivity mapping. PIGMENT estimates demonstrated strong biological validity, preserving submillimeter cortical microarchitectural patterns and early-childhood white matter developmental trajectories from 10-fold accelerated scans. Furthermore, PIGMENT enables reliable quantitative tensor mapping on cost-efficient low-field systems and the extraction of tumor-related biomarkers using ultra-fast clinical protocols. Together, these results establish PIGMENT as a physics-informed foundation model that extends quantitative diffusion MRI into regimes traditionally too sparse, heterogeneous, or clinically constrained for reliable analysis.

Physics-Informed Neural Networks (PINNs) are a common class of machine learning-based partial differential equation (PDE) solvers which train a network to represent a solution by minimizing a residual loss that encodes the PDE. Despite their successes, they are known to fail on certain simple equations, converging to an incorrect solution despite low loss. These failure modes have garnered significant attention in the literature over the past several years, motivating both architectural and optimization based solutions. By directly visualizing the residual, we show that failure modes are the result of overfitting: the loss is minimized on the collocation points, but not elsewhere. Applying regularization causes the failure modes to vanish. Finally, we extend double backpropagation over the full set of residuals, and use it to achieve state-of-the-art performance on four standard failure mode equations with up to $23\times$ fewer collocation points and a vanilla architecture.

Cross-Reynolds generalisation in neural PDE solvers remains poorly characterised. On the canonical forced 2D Navier-Stokes benchmark, a trained Fourier Neural Operator reaches 46.68% relative L2 error under a 10x Reynolds-number shift, yet zero-forward-model retrieval baselines already improve to 41-42%. This suggests representation geometry as a major organising variable among the tested methods. We test this hypothesis through ConvAE-Relay, which matches states in a source-trained convolutional autoencoder latent space and borrows dynamics from a source-regime database, achieving 38.34+/-0.07% using only a source-regime database and no target-regime fitting, labels, or database entries. A 2x2 ablation isolates matching quality as dominant over the update rule. Oracle experiments confirm that source-regime dynamics directions remain transferable (cosine similarity ~0.84) when matching stays on-manifold; autoregressive drift is the primary bottleneck (~12 percentage points). From the learned-prediction side, a U-Net with multi-scale skip connections achieves 34.72+/-0.60%, consistent with the retrieval-side finding that local, multi-scale representations organise cross-Reynolds transfer among tested methods. All claims are scoped to this benchmark.

This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear combination of basis functions integrated into the network, with coefficients determined by a direct least-squares solve, thereby bypassing traditional gradient-based training. The key methodological contribution include: (1) an efficient tensor-product scheme that generates multi-dimensional basis functions from combinations of two sets of subnetwork outputs, significantly reducing model complexity and parameter count while maintaining expressivity; (2) a block time-marching strategy to improve computational efficiency in long-time simulations; and (3) a linear reformulation strategy for handling nonlinear PDEs by treating known nonlinear terms as sources. TPNet achieves superior accuracy and shorter training times than conventional neural network solvers. This performance gain stems from its structured design and deterministic least-squares fitting, which contrast with the iterative, often computationally intensive optimization required by mainstream methods like Physics-Informed Neural Networks (PINNs).

We develop a comprehensive mathematical and computational framework for sequential surrogate modeling of three-phase black-oil reservoir dynamics using neural operators, with particular emphasis on Fourier Neural Operators (FNO) and their physics-informed variant (PINO). The application focus is the Norne benchmark reservoir, defined on a heterogeneous $46\times112\times22$ grid ($N=113,344$ cells), with a production history spanning $T=30$ timesteps covering 3298 days. Our theoretical contributions are organized around four interlocking problems: (1) functional-analytic formulation in a product-Sobolev-space setting, including well-posedness of the implicit timestep map and sharp local Lipschitz estimates; (2) covariate shift quantification, proving that the Wasserstein-2 distance grows as $W_2 \leq \varepsilon(L^n-1)/(L-1)$, with exponential population-risk discrepancy for $L>1$; (3) physics-constrained spectral stability, showing PINO training with $λ_R \geq λ^*_R$ reduces the learned Jacobian spectral radius to $ρ_F + Cλ_R^{-1/2}$, yielding uniform-in-time rollout error $|δ_n| \leq \varepsilon/(1-ρ)$; and (4) $K$-step TBPTT gradient analysis, deriving geometric bias decay $O(ρ^K)$, optimal window $K^ = O(\log(T/σ^2))$, and Adam convergence $O(1/\sqrt{t}) + O(ρ^{K^*})$. Empirical validation confirms all theoretical predictions: autoregressive PINO surrogates sustain $R^2>0.99$ (oil), $R^2>0.90$ (gas), $R^2\approx 0.80$ (pressure), and monotonically improving $R^2$ (water) across the full 3298-day horizon, trained on eight NVIDIA B200 GPUs in under one hour. A 1000-member ensemble runs in under one minute on a single B200 GPU, giving a ${\sim}10^4\times$ wall-clock speedup over the OPM finite-volume simulator.

Autonomous space systems operating in extreme thermal environments require accurate and efficient thermal modeling to support both pre-mission system design and onboard autonomy. For lunar rovers, large temperature gradients, radiative heat transfer, and variable surface conditions make reliable thermal prediction especially challenging. High-fidelity physics-based simulations provide accurate results but are computationally expensive, while simplified models and lookup-table approach often lack sufficient accuracy. Physics-informed machine learning (PIML) offers a promising alternative by combining data-driven models with embedded physical knowledge. This paper presents a PIML framework for thermal analysis of a simplified lunar rover with internal heat sources, where machine learning enables environment-adaptive coarse meshing. The proposed architecture integrates a transfer neural network (TNN) that adaptively determines 3D finite-difference nodalization based on thermal loads and initial conditions, enabling more accurate coarse-mesh calculations. A differentiable finite-difference thermal simulator is embedded within the framework to enforce physical consistency and support efficient training, while an upscaling layer reconstructs high-resolution temperature fields from the coarse-grid solution. The proposed PIML approach is evaluated against high-fidelity fine-mesh simulations, low-fidelity fixed coarse-mesh models, and a purely data-driven artificial neural network (ANN). Results show that the PIML framework improves prediction accuracy by 50% and 39% relative to the coarse-mesh physics model and ANN model, respectively, while maintaining physically consistent thermal distributions. Computationally, the framework is also 3x faster than high-fidelity simulations, demonstrating an effective balance between accuracy and efficiency for thermal modeling of lunar rover systems.

We propose KO-PDE-IDENT, a data-driven framework for identifying parsimonious partial differential equations (PDEs) with false discovery rate (FDR) control. PDE discovery from noisy observations is often hindered by extreme multicollinearity among candidate terms, which causes typical sparse-regression methods to select spurious terms. To address this problem, KO-PDE-IDENT initially mines a support set of potential candidate terms via model-X knockoff filters with finite-sample FDR control, then refines and ranks the surviving PDE alternatives. The framework integrates three components. First, knockoff feature statistics are constructed by coupling $\ell_{0}$-constrained adaptive best-subset selection with SHapley Additive exPlanations (SHAP), yielding an effective and computationally efficient difference statistic. Second, a recursive feature elimination (RFE) procedure removes terms whose marginal contributions are dispensable and assesses statistical necessity through knockoff-perturbed hypothesis testing. Third, the final model selection is formulated as a multi-criteria decision-making (MCDM) problem, where the optimal governing equation is the alternative that best balances a wide range of criteria such as predictive accuracy, model complexity and coefficient uncertainty. We evaluate KO-PDE-IDENT on five canonical PDEs under severe noise corruption. Empirical results show that our framework can exactly recover the true PDE structure, eliminating false discoveries while retaining all true underlying terms, with low coefficient estimation error.

Reconstructing continuous state trajectories of chaotic dynamical systems from sparse, noisy observations remains a fundamental open problem in nonlinear science. We introduce the Physics-Informed Diffusion Model with Dormand-Prince Integration (PIDM-DP), which embeds a fully differentiable 5th-order Dormand-Prince (DP-RK45) ODE integrator directly into the reverse sampling loop of a Denoising Diffusion Probabilistic Model (DDPM). At each denoising step, physics residuals are back-propagated via automatic differentiation, constraining every generated trajectory to satisfy the system's governing equations to 5th-order accuracy. A linear-scheduled guidance mechanism that ramps the physics weight from zero at high noise levels to its full value near the clean-data limit prevents the gradient explosions that cause naive physics-informed approaches to fail on stiff systems with Jacobian eigenvalues of order $O(10^3)$. Evaluated across five benchmark systems of increasing complexity 3D Lorenz, 3D Rössler, 5D Hyperchaotic, 20D Lorenz-96, and the stiff 3D Rabinovich-Fabrikant at 10% observation density with additive Gaussian noise ($σ=0.05$), PIDM-DP achieves reconstruction RMSE improvements of up to $15.4\times$ over an unconstrained diffusion baseline and decisively outperforms the Ensemble Kalman Filter on stiff systems where ensemble covariance collapses. On the Rabinovich-Fabrikant out-of-distribution benchmark, PIDM-DP attains RMSE $0.1097 \pm 0.0269$ versus $0.9443 \pm 0.5288$ (unconstrained diffusion, $8.6\times$ worse) and $0.3561 \pm 0.3040$ (EnKF, $3.2\times$ worse), with $p<0.001$ in paired Wilcoxon tests ($N = 30$). Topological validation via the Rosenstein Lyapunov estimator confirms that PIDM-DP preserves the chaotic invariant measure.

Starting from the axiomatic formulation of scale-space theory, we develop a viscosity-solution framework for multiscale image representations arising from degenerate elliptic-parabolic partial differential equations. Rather than introducing a new semigroup theory, we work within the standard viscosity-solution setting, using comparison principles to obtain well-posedness, uniqueness, and contraction in the supremum norm. This perspective is used to motivate a hybrid reconstruction operator in which a learned inverse map is followed by a nonlinear diffusion evolution. At the continuous level, the diffusion operator satisfies non-expansiveness, which provides stability for the reconstruction process; this framework is then evaluated on a CT-based mesothelioma classification task, where it attains an AUC of 0.875 with negligible variation across epochs, while the baseline model acquires AUC values from 0.49 to 0.80 without a clear convergence pattern. These observations are consistent with the stabilizing role suggested by the discussed viscosity theory.

Physics-informed machine learning (PIML) integrates mechanistic knowledge, typically in the form of partial differential equations (PDE), into data-driven models. Despite strong empirical performance, its statistical generalisation properties remain poorly understood, particularly in the regression setting with unbounded losses. Existing analyses rely on approximation or stability arguments and do not fully capture how physical structure influences generalisation from finite data. In this work, we develop a PAC-Bayesian framework for PIML that provides high-probability generalisation guarantees in the presence of unbounded losses. We adopt a multi-task perspective that jointly treats data fidelity, PDE residuals, initial and boundary conditions, avoiding the looseness induced by standard union-bound approaches. Our analysis leverages the structure of physics-informed objectives to derive novel bounds where the complexity scales with input-gradient norms of the losses, revealing a direct link between physical regularity and generalisation. We instantiate this framework under Sobolev and Poincaré-type assumptions, yielding two classes of bounds that trade off statistical complexity and smoothness in different regimes. Building on these results, we propose a self-bounding-aware learning algorithm that directly optimises tractable surrogates of the derived bounds, along with a practical procedure to estimate the associated constants in realistic settings. Empirical evaluations on standard PDE benchmarks demonstrate that our bounds are non-vacuous, significantly tighter than union-bound baselines, and can be effectively minimised during training. Overall, our results provide a principled statistical foundation for the generalisation of physics-informed models.

A recent conjecture by Joel Fine posits a relationship between the coefficients of the HOMFLY polynomial of a knot $K$ in the 3-sphere $S^3$, and the signed count of minimal surfaces in hyperbolic 4-space $\mathrm{H}^4$ meeting the sphere at infinity at $K$, with prescribed genus and self-intersection number. In this paper, we develop a novel machine learning framework based on Physics-Informed Neural Networks (PINNs) to solve the minimal surface equation in hyperbolic space. We utilise this framework to test Fine's Conjecture by constructing near-minimal surfaces bounding various families of knots in $S^3$. Furthermore, we develop an algorithmic method to find self-intersections and compute their sign. For every knot analysed, the computationally discovered minimal surfaces and their self-intersection numbers perfectly align with the predictions of Fine's Conjecture, providing empirical evidence for it.

Neural PDE solvers have followed the scaling trajectory of vision and language, with recent foundation models reaching billions of parameters. We argue that scale is a poor substitute for architectural inductive bias in this domain: structured priors deliver outsized parameter efficiency, and the pattern of where they succeed and fail is itself informative about what they capture. We instantiate this argument in WaveLiT, an architecture combining a discrete wavelet transform for lossless multi-resolution tokenization, an augmented linear attention block, a shared-weight multiscale feature pyramid, and a wavelet-domain auxiliary loss. Bespoke 1-10M-parameter WaveLiT models compete with foundation models of 100-1000$\times$ their size across eight TheWell benchmarks, with the largest gains on wave and acoustic-dominated benchmarks where the wavelet-multiscale prior fits the dominant dynamical structure and small per-step errors do not compound geometrically under rollout. Trained jointly across all eight benchmarks, a 10M-parameter foundation variant exhibits a structured, physically interpretable transfer pattern -- strongest where the wavelet-multiscale prior matches the dynamics, weakest on chaotic advection-dominated flows. The entire pipeline trains on a single GPU. The results suggest that small-model PDE performance is shaped by architectural inductive bias rather than scale, and that the structure of a prior's failures is a useful empirical signal about its content.

Accurate reconstruction of magnetic fields in inaccessible regions is vital for many high-precision experiments in physics. Traditional methods, such as spherical harmonic expansion, often suffer from truncation errors that limit their precision. This study proposes an advanced Physics-Informed Neural Network (PINN) framework for high-precision 3D magnetic field mapping. Unlike conventional data-driven models, the proposed PINN integrates Maxwell's equations directly into the loss function, enforcing divergence-free and curl-free conditions across the entire domain. A key innovation is the inclusion of explicit physics-residual losses at measurement locations, ensuring rigorous physical consistency beyond random collocation sampling. Validation using simulated data achieves a reconstruction accuracy of $10^{-4}$, a tenfold improvement over existing PINN benchmarks. Furthermore, experimental validation using a custom coil assembly demonstrates robust reconstruction with sub-percent relative accuracy, reaching the $10^{-3}$ level under ambient conditions. This AI-driven methodology provides a robust, high-precision solution for field monitoring and measurement in complex experimental environments where direct sensor placement is restricted.

Reconstructing PDE solutions from sparse observations is a core challenge in scientific computing. We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients (or initial states) and solutions (or final states), enabling both forward simulation and inverse recovery with limited paired data. At inference, sampling is guided by a composite loss that enforces agreement with sparse measurements and reduces the PDE residual; we support deterministic, stochastic, and hybrid samplers. We provide error guarantees for these guided procedures. For the deterministic optimizer, a coercivity condition ensures trajectory boundedness and a phase-wise contraction yields logarithmic complexity in the target accuracy. For the stochastic sampler, we introduce adaptive guidance and assume dissipativity of the velocity field to obtain uniform moment bounds independent of the noise-floor parameter. This leads to polynomial-time error bounds, and a matching lower bound shows constant guidance induces an unavoidable positive bias, motivating adaptivity. A hybrid deterministic-stochastic analysis is also provided. Experiments on static and time-dependent benchmark PDEs demonstrate competitive accuracy and faster inference than diffusion-based generative models.

Neural operators learn mappings from function-dependent inputs to solutions, providing an effective framework for solving partial differential equations (PDEs). For time-dependent PDEs, existing methods typically perform long-horizon prediction through autoregressive rollout directly in high-dimensional physical field spaces, where each predicted state is recursively fed back as the input for the next step. Although effective for short-term prediction, this autoregressive rollout and the lack of continuous-time modeling lead to progressive error accumulation over long-horizon rollouts. In this work, we propose Autoregression-Free Neural Operators (AFNO), which map the time evolution of PDEs into a latent space and model continuous-time vector fields within it. AFNO uses flow matching to learn the latent vector field, thereby enabling continuous evolution over extended horizons, avoiding autoregressive rollout and capturing dynamics under varying parameter configurations through explicit conditioning on physical parameters. Theoretical analysis and extensive experiments on six PDEs demonstrate that AFNO improves long-horizon prediction stability and consistently reduces rollout errors compared with the baselines.

Inverse problems in partial differential equations (PDEs) involve estimating the physical parameters of a system from observed spatiotemporal solution fields. Neural networks are well-suited for PDE parameter estimation due to their capability to model function-to-function space transformations. While existing benchmarks of machine learning methods for PDEs primarily focus on the forward problem, there are no similar comprehensive studies and benchmark datasets on PDE inverse problems, i.e., mapping solution fields to underlying physical parameters. We fill this gap by introducing PDEInvBench, a comprehensive benchmark dataset consisting of numerical simulations for both time-dependent and time-independent PDEs across a wide range of physical behaviors and parameters. Our dataset includes evaluation splits that assess performance in both in-distribution and various out-of-distribution settings. Using our benchmark dataset, we comprehensively explore the design space of neural networks for PDE inverse problems along three key dimensions: (1) optimization procedures, analyzing the role of supervised, self-supervised, and test-time training objectives on performance, (2) problem representations, where we study the value of architectural choices with different inductive biases and various conditioning strategies, and (3) scaling, which we perform with respect to both model and data size. Our experiments reveal several practical insights: 1) neural networks perform best with a two-stage training procedure: initial supervision with PDE parameters followed by test-time fine-tuning using the PDE residual, 2) incorporating PDE derivatives as input features consistently improves accuracy, and 3) increasing the diversity of initial conditions in the training data yields greater performance gains than expanding the range of PDE parameters. We make our dataset and codebase publicly available.

We introduce "Courant", a Perceiver-based encoder-processor-decoder surrogate model that has latent features exhibiting adaptive specialization and local support in the physical space, enabling functionality akin to an adaptive hp-refinement scheme, an attribute that is highly desirable in traditional numerical solvers and scientific machine learning broadly. The proposed architecture combines a shared random Fourier feature coordinate embedding, state-adapted latent queries, and a light-weight decoder. Courant is trained end-to-end with steady or transient simulation data and only a standard L_2 prediction loss in the physical space, achieving competitive accuracy on benchmarks. We demonstrate that Courant's inductive biases yield latents that are interpretable by design: they develop multiscale geometric specialization in the simulation domain and track coherent structures in the time-dependent case, acting analogously to time-evolving spatial basis functions and allowing for decoding a compact, geometry-anchored, partition-of-unity-like decomposition of the simulated field.

Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN expressivity for learning solutions to non-linear partial differential equations (PDEs). Despite their widespread use in practical applications, a rigorous theoretical understanding of the approximation properties of RaNNs in this context remains limited. Here, we derive error bounds for RaNN approximations to time-dependent Sobolev functions and obtain a dimension-free approximation rate $\frac{1}{2}$ for sufficiently regular functions. We apply our results to two important classes of non-linear PDEs: Porous Medium Equations and Compressible Navier-Stokes Equations, showing that RaNNs are capable of efficiently approximating solutions to these complex, non-linear PDEs. Our theoretical analysis is supported by numerical experiments, showing that the obtained convergence rates extend beyond the considered setting.

While Physics-Informed Neural Networks (PINNs) are powerful for solving Partial Differential Equations (PDEs), their training is often paralyzed by gradient pathology. The gradients from the PDE residuals and boundary constraints oppose each other, trapping the model in local minima. Current solutions, such as adaptive weighting or hard constraints, either fail to fundamentally resolve this ill-conditioning or are limited to simple geometries. In this study, we systematically analyze the possible causes of this gradient pathology from the perspectives of loss landscapes and optimization dynamics. Based on the obtained conclusion, we propose Constraint-Aligned loss with Manifold Lifting (CAML). By reformulating all zeroth-order terms into aligned constraints, our method effectively mitigates gradient conflicts. In addition, we introduce a delay factor to help the optimizer skip the high-curvature area. Experiments demonstrate that our CAML significantly enhances numerical stability and efficiency in highly complex PINN problems. Our code is open-sourced on https://github.com/YichenLuo-0/CAML.

We introduce a novel neural operator architecture designed to approximate solutions of linear elliptic partial differential equations with high-contrast, spatially varying coefficients. The network, termed the Iterated V-shaped Net (IV-Net), realizes a mapping from the input coefficients and righthand side to the corresponding solution field. The architecture of IV-Net is informed by, and closely resembles, a V-cycle multigrid solver. The IV-Net model is parameterized via convolutional layers defined in the physical domain. For coercive problems with highly heterogeneous coefficients, the proposed network exhibits superior performance relative to a proper orthogonal decomposition (POD) approach and several existing neural operator architectures. For low-frequency oscillatory Helmholtz problems with smooth coefficients, its performance is similar to that of a Fourier neural operator. We analyze the approximation error and convergence behavior of IV-Net, its data efficiency, and its dependence on the underlying discretization mesh. Furthermore, we demonstrate the practical effectiveness of the architecture through a series of numerical experiments, including applications to uncertainty quantification, inverse problems, and prediction of quantities of interest.

Advanced driver assistance systems (ADAS) play an important role in modern automotive intelligence, significantly enhancing vehicle safety and stability. The performance of ADAS critically relies on accurate and reliable vehicle state estimation, particularly from vehicle dynamic sensors. Among these signals, wheel load is a key variable for chassis control and safety-critical functions, yet it remains difficult to estimate robustly due to complex suspension geometry, nonlinear dynamics, and measurement noise. To address this issue, we propose DBPnet, a Bayesian physics-informed neural network (PINN) with a physics-aware embedding module inspired by damper characteristics. First, this paper presents a suspension linkage-level modeling (SLLM) approach that constructs a nonlinear instantaneous dynamic model by explicitly considering the complex geometric structure of the suspension. Building upon SLLM, Bayesian inference is integrated into the PINN to effectively cope with noise and uncertainty in the vehicle chassis system, thereby improving the model's robustness. Then, a physics-informed loss function is employed to ensure consistency with fundamental physical principles, while the damper characteristics-inspired embedding module extracts temporal variation features of input signals and incorporates them into each layer of the PINN, ensuring that physical observations guide the neural network without being constrained by fixed physical models. Extensive evaluations on high-fidelity simulations and real-world experiments demonstrate that our DBPnet consistently achieves lower RMSE and MaxError than baseline methods. These results highlight the potential of our DBPnet to advance wheel load estimation and contribute to the development of more reliable ADAS actuator functions.

We propose Hermite-NGP, a gradient-augmented multi-resolution hash encoding designed to enable fast and accurate computation of spatial derivatives for neural PDE solvers. Unlike existing NGP-based approaches that rely on automatic differentiation or finite differences and suffer from instability or high cost, Hermite-NGP explicitly stores function values and mixed partial derivatives at hash grid vertices, allowing fully analytic evaluation of gradients, Jacobians, and Hessians via Hermite interpolation. This design preserves the efficiency and spatial adaptivity of NGP while supporting analytic differential operators up to second order. We further introduce a multi-resolution curriculum training strategy analogous to multigrid V-cycles to enable coarse-to-fine optimization. Across a range of 2D and 3D PDE benchmarks, Hermite-NGP achieves up to approximately 20 times lower error than prior neural PDE methods, and reduces wall-clock convergence time by 2 to 10 times compared to other solvers, with per-epoch training times as low as 3.5 ms for models with up to 17M parameters.

This letter introduces a physics-informed self-supervised framework for sonar image despeckling that reformulates despeckling as residual consistency in the homomorphic log domain. By constraining the log-ratio residual to obey multiplicative speckle statistics, the proposed method eliminates the need for clean supervision while preventing degenerate identity solutions. A variance-targeted statistical loss combined with edge-aware structural regularization and median-guided curriculum stabilization enables effective speckle suppression with preserved structural fidelity. This formulation along with a lightweight neural network achieves state-of-the-art performance across multiple real sonar datasets and demonstrates excellent cross-dataset robustness, while remaining suitable for real-time deployment.

We present a probabilistic modeling framework for incorporating small-scale spatial heterogeneity into macroscopic descriptions of material behavior for polycrystalline metallic materials. Spatially heterogeneous material state fields are represented using probability density functions (PDFs), providing a principled statistical description of microstructural variability and state evolution across different computational polycrystalline realizations. The framework is built on the inverse identification of a probabilistic transport model, formulated as a Liouville equation with an unknown drift term. To enable accurate, stable, and interpretable inference of this drift field in high-dimensional, transport-dominated settings, we develop a Separable Probability Learning Technique via Physics-Informed Neural Networks (SPLIT-PINN). This method incorporates a marginal-correction drift decomposition, orthogonality constraints, and residual-based adaptive training to enhance well-posedness, numerical stability, and physical consistency without imposing restrictive parametric assumptions. Using SPLIT-PINN, the drift field governing the temporal evolution of joint state PDFs is inferred directly from data. After benchmark validation, the framework is applied to physical computational datasets describing the evolution of polycrystalline microstructural states, including von Mises stress, dislocation density, and equivalent plastic strain rate. The learned Liouville model, trained on a single dataset, is subsequently used in forward predictions of the temporal evolution of joint and marginal PDFs for multiple unseen polycrystal realizations. Quantitative comparisons with reference PDFs demonstrate that the proposed framework yields accurate and robust probabilistic predictions and generalizes effectively across datasets.

This work introduces the Wavelet-Laplace Neural Operator (WLNO), a novel neural operator that fuses Haar wavelet multi-scale spatial decomposition with the Laplace-domain pole-residue formulation of the Laplace Neural Operator (LNO). While LNO captures transient and steady-state dynamics through learnable system poles and residues, it lacks an explicit mechanism for extracting spatially localized multi-scale features inherent in complex PDE solutions. WLNO addresses this by augmenting the LNO core with a parallel single-level Haar discrete wavelet transform (DWT) branch that decomposes the lifted feature map into four frequency subbands: approximation (LL), horizontal detail (LH), vertical detail (HL), and diagonal detail (HH) and applies independent learned $1\times1$ convolutions to each subband before reconstruction via the inverse DWT. The two branches are fused through a learnable sigmoid-gated weight $α_\mathrm{wav}$, initialized to give a small initial contribution to the wavelet branch, allowing the model to adaptively balance Laplace-domain dynamics against spatial multi-scale features throughout training. WLNO is evaluated against LNO on five benchmark PDE problems using identical hyperparameters, training data, and evaluation protocols: the diffusion equation, the Burgers equation, the reaction-diffusion system, Darcy flow, and the two-dimensional Navier-Stokes equation. WLNO consistently outperforms LNO on all five problems, with the most pronounced improvement on problems with strong spatial multi-scale structure, such as the Burgers equation with sharp shock fronts and the Navier-Stokes equation with coherent vortical structures, while remaining consistent across smoother and elliptic problems. These results demonstrate that wavelet-based multi-scale spatial decomposition is a principled and effective complement to Laplace-domain operator learning.

We propose a Weak-form Physics-Informed Neural Operator (WINO), a data-free framework that combines the efficiency of neural operators with the geometric flexibility of the $\varphi$-finite element method ($\varphi$-FEM). $\varphi$-FEM is an unfitted method that accommodates geometric variations without body-fitted meshes, where the domain geometry is represented by the level-set function $\varphi$. To impose the boundary conditions, Dirichlet problems adopt the $\varphi$-FEM lifting so only the homogeneous displacement contribution is learned, whereas traction-driven Neumann problems additionally predict the auxiliary fields necessary for the unfitted weak formulation. Parameters are trained by minimizing squared weak-form residuals aligned with $\varphi$-FEM together with squared penalties on the cut-cell auxiliary equations, which removes the need for large paired datasets of converged reference solutions. After training, WINO outputs can seed the nonlinear $\varphi$-FEM solvers as neural operator warm starts (NOWS), which reduce iteration counts relative to traditional cold-started solvers. Numerical benchmarks show that WINO achieves high accuracy below 0.04 across all benchmarks, while reducing total computational time by 50--80% compared with purely data-driven methods.

We present an improved neural field architecture for solving partial differential equations (PDEs). Current physics-informed neural networks (PINNs) provide a flexible framework for solving PDEs, but they struggle to achieve highly accurate solutions and require computation that scales poorly with parameter count. Our model, which we call beignet (Bandlimited Embedding with Interpolated Grid Network), replaces the random Fourier feature embedding used by existing PINN models with a trainable multi-resolution Fourier feature pyramid. To query beignet at a continuous coordinate, we use Fourier interpolation at each level of the pyramid to return features at the input coordinate, and then decode this vector with a fully-connected neural network trunk. Our model provides multiple benefits: 1) Spatial derivatives can be computed efficiently by using the chain rule to compose derivatives of the neural network computed with automatic differentiation with derivatives of the feature grid computed spectrally by the Fast Fourier transform (FFT). 2) beignet can achieve higher accuracy in a compute-efficient manner by scaling the parameter count of this Fourier feature pyramid, instead of the less-efficient strategy of scaling the neural network architecture. 3) beignet can directly control the representation bandlimit, resulting in more stable optimization for difficult PDEs. We demonstrate that beignet finds significantly more accurate solutions on PDE benchmarks using fewer parameters than state-of-the-art PINN methods. We further evaluate beignet on the self-similar inviscid Burgers blowup problem and show that it can minimize residuals to near machine precision using Adam, an accuracy regime previously attained only by using computationally expensive higher-order optimizers.

Rapid and accurate flood extent mapping from Remote Sensing data, such as Synthetic Aperture Radar (SAR), is critical for operational disaster response, but standard Deep Learning models often produce physically impossible predictions due to a lack of hydrological constraints. While PhysicsInformed Neural Networks (PINNs) attempt to address this by embedding governing laws directly into the loss function, their application to real-world remote sensing data frequently fails. Enforcing rigid spatial derivatives (e.g., the 2D Shallow Water Equations) onto unconditioned latent spaces attempting to fit noisy SAR speckle causes catastrophic gradient divergence, a phenomenon we term Physics Shock. In this paper, we propose a novel Uncertainty-Aware PINN framework tailored specifically for applied Earth Observation that addresses this instability. By integrating a dynamic Warm-Start protocol and modeling heteroscedastic aleatoric uncertainty via a negative log-likelihood objective, the network learns to dynamically relax physical constraints in regions of high sensor noise while strictly enforcing them in high-confidence areas. Evaluated on the Sen1Floods11 dataset, our probabilistic Attention-Gated FNO-UNet successfully stabilizes multi-objective optimization, achieving a +25% relative improvement in Intersection over Union (IoU) compared to deterministic baselines. Furthermore, through Deep Ensembles, we successfully disentangle intrinsic sensor noise from out-of-distribution terrain ignorance, providing operational agencies with highly calibrated, physically consistent confidence bounds for robust disaster mitigation and real-time decision-making.

The identification of constitutive neural network models from heterogeneous full-field deformation data provides a robust alternative to traditional calibration methods based on homogeneous stress-strain experiments, particularly given the high dimensionality of trainable parameters. Existing approaches must balance generality, robustness, and computational efficiency: Conventional finite element model updating is broadly applicable but computationally demanding; weak-form methods offer efficiency but are sensitive to noise and data scarcity; neural operator models are highly expressive but require extensive training datasets. This work presents FE-MAD (Finite Element-Based Material learning via Automatic Differentiation), an end-to-end differentiable framework that integrates a constitutive neural network model within a JAX-FEM nonlinear solver and identifies its parameters through gradient-based minimization of a measurement-mismatch loss. Newton tangent stiffness and loss gradients are computed automatically using forward- and reverse-mode automatic differentiation throughout the entire pipeline, thereby removing the need for analytic adjoints or offline surrogate models. FE-MAD is demonstrated for two architectures: a grey-box Constitutive Artificial Neural Network (CANN), a polyconvex, fully connected model with high flexibility, and a white-box CANN, an expert-system network with phenomenologically interpretable strain-energy terms. Focusing on incompressible isotropic hyperelasticity, FE-MAD is evaluated on three open experimental datasets: (1) full digital image correlation (DIC) of a perforated tensile specimen, (2) a reduced-data scenario with a one-dimensional stretch profile and global force-displacement curve, and (3) a heterogeneous matrix-inclusion system in which both phases constitutive laws are identified and generalized to twenty-two previously unseen samples.

When learning dynamical systems from data, embedding physical structure can constrain the solution space and improve generalization, but many physics-informed models assume access to the full system state. This limits their use in partially observed settings, where some state variables are completely unobserved and must be inferred without direct supervision. Here, we present neural Hamiltonian ordinary differential equations (NHODE), a framework that combines Hamiltonian neural networks (HNNs) with neural ordinary differential equations (neural ODEs) to learn partially observed dynamical systems from data. The Hamiltonian structure enforces energy conservation by construction, while the neural ODE framework enables a flexible training procedure that allows the loss to be defined only on observed variables. We also incorporate additional physical constraints through symmetry-aware coordinate transformations and separable energy formulations. The framework is evaluated on systems of increasing complexity, from linear and nonlinear mass-spring systems to the chaotic three-body problem. Across all examples, increasing the amount of embedded physical structure improves the accuracy and long-horizon stability of the predictions. Even in the most challenging regimes, the NHODE framework captures both observed and latent dynamics, whereas purely data-driven baselines become unstable.

Physics-informed neural networks (PINNs) for coupled multiphysics systems suffer systematic accuracy degradation as inter-equation coupling strengthens. We provide a theoretical explanation through neural tangent kernel (NTK) analysis: for linearly coupled systems, the standard NTK's spectral radius grows as $Ω(γ^2)$ with coupling strength $γ$, shrinking the stable learning rate, while block-diagonal Gauss--Newton (GN) preconditioning yields a preconditioned NTK $K_P = JH^{+}J^\top$ whose spectral radius is bounded by $S$ (number of networks), independent of $γ$. Adam's diagonal preconditioning destroys this projector structure -- inflating $λ_{\max}$ far above $S$ for any coupling type -- and its residual-dynamics kernel grows as $Θ(γ)$, placing its stable learning rate strictly between gradient descent and GN. For one-way coupling the limitation is class-wide: no diagonal preconditioner, fixed or adaptive, halves the driving residual in fewer than $Ω(γ)$ iterations ($Ω(γ^2)$ if fixed), whereas block-diagonal GN requires $O(1)$. We verify $Ω(γ^2)$ growth across linearly coupled benchmarks and confirm $λ_{\max}(K_P) = S$ in all three 1D systems, including nonlinearly coupled NP+P. Combining the Kronecker-preconditioned optimizer SOAP with inverse-gradient-norm loss balancing (SOAP+GradNorm) yields coupling-robust accuracy: across 222 experiments spanning three 1D systems and a 2D electroosmotic flow benchmark, SOAP+GradNorm maintains final-epoch $L_2$ accuracy across coupling strengths, with $\leq 2.3\times$ degradation in nonlinear NP+P while Adam+GradNorm fails ($L_2 > 0.1$). SOAP+GradNorm further scales to a 2D, 6-PDE electroosmotic flow at EDL-resolved conditions down to $\varepsilon = 0.01$ -- a regime all prior PINN electrokinetics studies have avoided -- where Adam+GradNorm fails entirely ($L_2 > 0.3$).

Active traffic management (ATM) is frequently hindered by traditional macroscopic models and rigid empirical thresholds that fail to capture metastable phase precursors, resulting in delayed, reactive interventions. To address this, we propose SpinFlow, a physics-informed spin-field framework unifying Kerner's three-phase theory with statistical physics for continuous macroscopic traffic phase inference. Inspired by the Heisenberg model, SpinFlow parametrizes spatially varying phase weights via a latent spin vector and a competitive-equilibrium mapping, allowing synchronized flow to emerge naturally. A physics-regularized Expectation-Maximization algorithm inverts this latent structure from high-resolution trajectories, jointly optimizing the spin field while softly enforcing mass conservation and spatial smoothness. We introduce the Phase Equilibrium Degree (PED) to quantify structural alignment and topologically localize phase-transition points. Across four real-world trajectory datasets, SpinFlow achieves $R_{q}^{2}$ up to 0.940, PED drops of 94.9-100%, and interpretable phase maps that outperform three heterogeneous baselines on forward accuracy, physics consistency, and bottleneck localization. SpinFlow pinpoints congestion nucleation without prior network topology, yielding a data-driven, physics-consistent trigger for ATM.

Defocus deblurring in pathological microscopy remains challenging due to the spatially varying and locally discontinuous nature of optical blur induced by a position-dependent integral imaging process. Existing deep learning methods, constrained by shift-invariance assumptions and limited interpretability, are not well suited to such heterogeneous blur patterns. Neural operators provide a principled alternative by modeling defocus formation directly as an integral operator, offering a new perspective on defocus deblurring. However, most existing neural operator architectures for low-level vision rely on globally parameterized kernels that assume smoothness and stationarity, limiting their ability to model heterogeneous and locally discontinuous blur patterns. To address this limitation, we propose the Discontinuous Galerkin Neural Operator (DGNO), which parameterizes the integral kernel using a discontinuous Galerkin formulation with element-local volume operators and interface numerical fluxes. DGNO provides a principled combination of locality, heterogeneity modeling, and global coherence while preserving the underlying physics of optical image formation. Extensive and insightful experiments demonstrate that DGNO surpasses state-of-the-arts, delivering sharper reconstructions, robust handling of spatially varying blur, and scalable high-resolution performance. The code will be released at https://github.com/DeepMed-Lab-ECNU/Single-Image-Deblur.

We introduce EMMA, a physics-informed multimodal framework that recovers all identifiable dynamical parameters of a system directly from raw video, audio, and image-based time-series observations. Unlike prior video-only approaches that struggle with occluded states, hidden actuation inputs, or assumptions about known initial conditions and coordinate frames, EMMA performs joint inference of explicit parameters, implicit dynamical components, and calibration invariants within a unified continuous-time model. EMMA leverages a Liquid Time-Constant (LTC) network to learn latent dynamics from heterogeneous modalities while a physics-constrained loss enforces consistency with the governing differential equations. A unified feature pipeline enables consistent alignment across video trajectories, acoustic signatures, and chart-derived measurements, allowing EMMA to estimate parameters under forced, implicit, and multivariate dynamics without requiring segmentation masks, differentiable rendering, or specialized sensors. Across 100+ scenarios including five standard dynamical benchmarks (75 Delfys videos), real-world rover and quadrotor systems with hidden inputs, and simulation-chart case studies spanning biological and chaotic systems, EMMA delivers robust multi-parameter recovery and significantly outperforms existing single-modality and equation-discovery baselines. Our results establish EMMA as a general, scalable solution for physics-consistent model extraction from opportunistic multimodal data. Code and data are available at: https://github.com/ImpactLabASU/EMMA-CVPR2026

Neural operators serve as fast, data-driven surrogates for scientific modeling but typically rely on a monolithic, single-pass inference procedure that struggles to resolve high-frequency details, a limitation known as spectral bias. We introduce the Iterative Refinement Neural Operator (IRNO), which augments pre-trained operators with a learned refinement module iteratively applied via fixed-point iteration. IRNO decomposes the prediction into a coarse initialization followed by successive residual corrections, paralleling classical numerical solvers. Under local assumptions, we establish contraction of the induced operator, ensuring convergence to a unique fixed point. To explicitly target high-frequency errors, we propose a progressive spectral loss that adaptively increases penalty on high-frequency components over refinement steps during training. Across physical systems, IRNO consistently lowers error, with up to 56.05% improvement on turbulent flow. On Active Matter, spectral analysis reveals that, relative to base operator, the normalized error ratios decrease to 27.72-36.10% in low-, 5.07-6.68% in mid-, and 1.48-2.04% in high-frequencies, remaining stable beyond the trained iteration count. Code is available at https://github.com/xiaotianliu-dartmouth/Iterative_Refinement_Neural_Operator

Digital elevation models (DEMs) underpin terrain analysis in Geographic Information Systems (GIS), but in their common raster form, they rely on interpolation for off-grid sampling and finite-difference operators for derivative-based analysis. Implicit neural representations (INRs) offer a continuous alternative, but prior terrain INRs lack explicit frequency control, neglect the gradient structure of terrain, and remain too large and costly to train for practical deployment. We present ImplicitTerrainV2, which advances terrain INRs toward a compact, efficient neural terrain data format by combining a spectral control mechanism with wavelet-guided spatial adaptivity, derivative-aware supervision, and post-training model compression. At its core, a wavelet complexity field (WCF) derives spatially-adaptive frequency masks from analytically computed wavelet coefficients, localizing high-frequency capacity to complex terrain regions. The same field guides complexity-aware adaptive sampling that concentrates training in high-complexity regions, while gradient matching applies extra supervision to enforce the smooth manifold structure of terrain DEMs for improved derivative fidelity. Post-training mixed-precision quantization and entropy coding reduce storage to 1.23 bpp with a 0.28 dB PSNR drop. On 50 Swiss terrain tiles, ImplicitTerrainV2 reaches 66.25 dB end-to-end PSNR, improving over the prior work by 5.70 dB while using 3.2x fewer parameters and training in 55 s per tile on a single GPU. Our compressed neural format is competitive with several established DEM codecs in rate-distortion performance, while additionally supporting off-grid point queries, closed-form derivative evaluation, and resolution-independent reconstruction, which may benefit many downstream GIS applications.

Partial differential equation (PDE) foundation models are pretrained networks that forecast how physical fields like velocity and pressure evolve from a single reusable solver. On unfamiliar flows their predictions drift step by step, errors concentrate in a few regions, yet retraining destabilizes the network and uniform post-hoc correction overlooks this spatial concentration. To address this, we propose a frozen-solver post-hoc correction framework, Adaptive Risk-Calibrated Spatial Triage for Auditable Refinement (ARC-STAR). ARC-STAR organizes correction into three stages: a global corrector removes broad solver bias, a blockwise local refiner cleans the post-global residual, and, at deployment, a label-free score routes refinement to high-risk blocks under a compute budget. The framework is designed to be (i) frozen-host, preserving the pretrained solver without fine-tuning; (ii) auditable, with global and local stages trained and evaluated separately for measurable contributions; and (iii) budget-aware, using a blockwise interface that either refines the full field or routes limited compute to high-risk regions. Across five flow benchmarks spanning ten regime cells, ARC-STAR is the only method that cuts velocity rollout error by at least 36x over raw Poseidon on every cell. The global stage reduces raw host error by 91-99%, and the local stage further reduces the remaining post-global residual by up to 94.4%.

Neural operators have achieved significant success in modern scientific computing due to their flexibility and strong generalization capabilities. Existing models, however, primarily rely on first-order kernel integral approximations, which severely limit their expressivity. To address this, we propose the Infinite-order Kernel Neural Operator (IKNO), which constructs neural operators via infinite-order kernel integrals and admits an elegant closed-form finite approximation. We develop two complementary infinite-order neural operator constructions: IKNO-Vanilla, which applies the full-kernel resolvent on the product grid via Kronecker eigendecomposition, and IKNO-TP, an alternative tensor-product operator that composes per-axis resolvents. Furthermore, we develop fast computation schemes for both variants of IKNO, which achieve outstanding global information aggregation while maintaining high computational efficiency. Empirically, we evaluate our IKNO on both time-dependent and time-independent benchmarks with arbitrary input shapes, including large-scale industrial datasets. Extensive experiments demonstrate that the IKNO method consistently achieves the SOTA accuracy with significant improvements on nearly all benchmark datasets while maintaining scalability to very large point clouds.

The integration of machine learning with domain-specific physics is transforming the design, monitoring, and control of electricity systems, where data scarcity, limited interpretability, and the need to enforce physical laws constrain purely data-driven models. Physics-informed machine learning (PIML) addresses these limitations by embedding governing equations directly into the learning process, yielding accurate, efficient, and scalable solutions for Industry 4.0 applications. This article reviews hybrid PIML architectures for electricity systems, including physics-informed neural networks (PINNs), Deep Operator Networks (DeepONets), Fourier Neural Operators, Extreme Learning Machine-enhanced PINNs, graph-based PINNs (PIGNNs), and domain-decomposition PINNs. Each approach is examined through case studies spanning field analysis, fault detection, digital twins, surrogate modeling, and control optimization. The review shows that embedding Maxwell's equations and other first-principles constraints substantially improves predictive accuracy under sparse and noisy data, reduces simulation time by orders of magnitude relative to finite element methods, and enhances generalization across operating regimes. Hybrid frameworks consistently outperform purely data-driven baselines on parameter sensitivity, dynamic behavior, and robustness, while supporting real-time digital-twin calibration and uncertainty quantification. Persistent challenges include training instability for stiff multi-scale problems, computational cost of high-fidelity models, and the absence of standardized benchmarks. The findings demonstrate that PIML enables a paradigm shift from black-box data-driven methods to transparent, physics-informed strategies, positioning the field for sustained innovation in resilient and intelligent electricity systems.

Physics-informed diffusion models typically enforce PDE constraints only on final outputs, leaving intermediate representations unconstrained and prone to shortcut learning under shifted boundary conditions. We introduce REPA-P, a teacher-free, architecture-agnostic framework that aligns intermediate features with physical states using first-principles residuals. REPA-P attaches lightweight $1{\times}1$ projection heads to selected layers, decodes hidden activations into physical quantities, and applies PDE residual losses during training. These heads are discarded at inference, introducing zero overhead. Across four PDE tasks, including Darcy flow, topology optimization, electrostatic potential, and turbulent channel flow, REPA-P accelerates convergence by up to $2{\times}$, reduces physics residuals by up to $66.4\%$, and improves out-of-distribution robustness by up to $49.3\%$, with consistent gains on both U-Net and Diffusion Transformer backbones. Ablations show that supervising a small set of intermediate layers captures most benefits and complements output-level physics losses. Code is available at https://github.com/Hxxxz0/REPA-P.

Accurate classification of breast ultrasound images into benign, malignant, and normal categories is a critical clinical task complicated by speckle noise, acoustic shadowing, and inter-class visual ambiguity. Existing deep learning methods rely on single-stream architectures with generic augmentation that ignores ultrasound acquisition physics, and no prior method dedicates a stream to the lesion boundary features identified as the most diagnostically significant visual cue. We propose HADS-Net, a Hybrid Attention-Augmented Dual-Stream Network exploiting global texture and local boundary cues through two parallel pathways. Stream 1 applies physics-informed augmentation simulating speckle noise, acoustic shadowing, and gain variation before extracting features via pretrained EfficientNet-B3 projected to 512 dimensions. Stream 2 extracts Sobel edge maps processed by a lightweight CNN projected to the same 512-dimensional space. A cross-attention fusion module allows the texture stream to selectively query boundary features, producing a jointly optimised representation classified by an MLP trained with adaptive class-weighted focal loss. Five-fold stratified cross-validation with cosine annealing over 50 epochs is used, with the globally best checkpoint selected by lowest validation loss evaluated on a held-out test set. On the BUSI dataset, HADS-Net achieves 96.58% accuracy, macro ROC-AUC of 0.9978, macro F1 of 0.9654, and per-class F1-scores of 0.970, 0.951, and 0.976 for benign, malignant, and normal. No malignant lesion is misclassified as normal. These results confirm that modality-specific augmentation with cross-modal attention fusion is an effective strategy for ultrasound-based breast cancer diagnosis.

JEPAs often regularize one-view embeddings toward an isotropic Gaussian, implicitly baking Euclidean symmetry into the representation. We show that this is not merely a benign default. For a known structured downstream geometry $H\succ0$, the minimax and maximum-entropy covariance under a Hamiltonian energy budget is $(c/d)H^{-1}$, and Euclidean isotropy incurs a closed-form price of isotropy. More importantly, when the downstream geometry is unknown, no geometry-independent fixed marginal target is canonical: every fixed covariance shape can be maximally misaligned for some structured geometry. We further show that even oracle one-view marginals do not identify the JEPA view-to-view predictive coupling. These results suggest that the structural bias in JEPAs should enter the cross-view coupling rather than a fixed encoder marginal. We instantiate this principle with \textbf{HamJEPA}, which encodes each view as a phase-space state $(q,p)$ and predicts view-to-view transitions with a learned Hamiltonian leapfrog map, while non-isotropic scale and spectral floors prevent collapse. In a deliberately headless token protocol, HamJEPA improves over SIGReg on CIFAR-100 by $+4.89$ kNN@20 and $+3.52$ linear-probe points at 30 epochs, and by $+6.45$ kNN@20 and $+10.64$ linear-probe points at 80 epochs, while a matched MLP predictor ablation shows that the symplectic coupling is the ingredient driving the neighborhood-geometry gain. On ImageNet-100, HamJEPA-$q$ improves by $+4.82$ kNN@20 and $+7.52$ linear-probe points at 45 epochs.

Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing approaches typically approximate such features within continuous function spaces, often requiring increased model capacity and high-resolution data. In this work, we propose Cut-DeepONet, a two-stage training framework that explicitly models discontinuities while reducing learning complexity. Our approach reformulates the problem via a lifting strategy, partitioning the domain into smooth subregions while representing discontinuities as boundaries in a higher-dimensional space. This separation aligns the operator learning task with the inductive bias of neural networks and avoids directly approximating discontinuities. An additional network predicts input-dependent discontinuity locations for unseen inputs, which are then used to guide the neural operator in generating smooth components within each region. Experiments on benchmark PDEs show that Cut-DeepONet outperforms state-of-the-art methods, even when trained on low-resolution datasets. The method excels on problems with discontinuities and sharp transitions, while using fewer trainable parameters. Our results highlight the benefits of changing the representation of operator learning rather than increasing model complexity.

This work presents an end-to-end strategy for solving inverse problems constrained by Partial Differential Equations within a fully differentiable Machine Learning framework. The proposed formulation provides a unified and user-friendly methodology applicable to a wide range of problems, from data assimilation to closure modeling. Our approach combines a baseline differentiable PDE solver, which predicts the state w from the nonlinear system $R(w) = 0$, with a generic additive, parametrized, and differentiable correction $f_φ(w)$, with trainable parameters $φ$. We show how to optimize phi within a fully differentiable Python workflow by reformulating the PDE as an implicit layer, enabling its integration into arbitrary objective functions, while leveraging PyTorch's automatic differentiation graph. The method is demonstrated on the Reynolds-Averaged Navier-Stokes equations for compressible flows, where the closure term, or a portion of it, is modeled using trainable parameters or a Neural Network. The first application considers the 2D NASA Wall-Mounted Hump test case, where a production-term parameter is optimized against time-averaged LES data. A second application is carried out on the VKI LS-59 turbine blade, where the Spalart-Allmaras eddy viscosity field is reconstructed through the optimization of a trainable spatial field. A dataset is generated starting from the VKI LS-59 turbine blade geometry using the differentiable BROADCAST solver with the Spalart-Allmaras turbulence model. The results highlight the flexibility of the framework, showing its applicability beyond turbulence modeling to a broader class of physics-informed PDE-constrained problems with data-driven components.

Physics-informed neural networks (PINNs) offer a mesh-free framework for solving partial differential equations (PDEs), yet training often suffers from gradient pathologies, spectral bias, and poor convergence, especially for problems with strong nonlinearity, sharp gradients, or multiscale features. We propose the Curriculum-Guided Gaussian Mixture Physics-Informed Neural Network (CGMPINN), which integrates Gaussian mixture modeling with dynamic curriculum learning. Specifically, a GMM is periodically fitted to the PDE residual distribution to quantify spatially varying learning difficulty. A smooth curriculum schedule progressively shifts training focus from easy to harder regions, while precision-based variance modulation suppresses unreliable clusters during early optimization. This dual curriculum is governed by a shared curriculum parameter and can be combined with self-adaptive loss balancing. We further establish theoretical guarantees, including sublinear convergence of the gradient norm for the induced time-varying loss, uniform equivalence between the curriculum-weighted and standard PDE losses, and a generalization bound with an explicit weighting-induced bias characterization. Experiments on six benchmark PDEs spanning elliptic, parabolic, hyperbolic, advection-dominated, and nonlinear reaction-diffusion types show that CGMPINN consistently achieves the lowest relative $L_2$ and maximum absolute errors among all compared methods, reducing relative $L_2$ error by up to 97.8% over the standard PINN at comparable cost. Our code is publicly available at https://github.com/Mathematics-Yang/CGMPINN.

Physical computing systems provide a promising route toward hardware-native machine learning, but their computational capabilities remain difficult to characterize in a principled, task-independent, and data-efficient way. We extend the Information Processing Capacity (IPC) framework to stationary physical computing systems and establish several fundamental results: individual capacities are bounded between zero and one, their sum over a complete basis is bounded by the number of readouts, and noise strictly reduces this bound. We address the finite-sample estimation of IPC and derive the asymptotic form of the systematic positive bias affecting naive estimators. Building on these results, we introduce data-efficient estimation methods based on Richardson extrapolation and Sobol quasi-random sampling. We validate the framework experimentally using a photonic computing system based on picosecond laser pulses propagating through a nonlinear optical fibre. By varying the laser power and fibre length, we observe systematic shifts of the IPC distribution toward higher-order nonlinear capacities induced by the Kerr effect. Finally, we demonstrate that the total IPC strongly correlates with performance on benchmark machine-learning tasks and provides a reliable estimate of the effective dimensionality of the system. These results establish IPC as a practical bridge between the intrinsic dynamics of physical computing systems and their machine-learning performance.

This study develops a dimensionless multi-domain physics-informed neural network (PINN) framework for electro-osmotic radial consolidation considering smear effects and combined vacuum and surcharge loading. Three PINN-based models are investigated: a standard soft-constrained PINN (Std-PINN), a modified gated PINN (Mod-PINN), and a modified gated PINN with hard-constraint boundary encoding (Mod-HC-PINN). The models are evaluated against FEM reference solutions under four loading cases, including constant vacuum, exponential vacuum, exponential vacuum with ramp surcharge, and exponential vacuum with cyclic haversine surcharge. The results indicate that the gated architecture applied in Mod-PINN improves the resolution of steep pressure gradients near the cathode and smear-zone interface under constant vacuum loading. Under time-dependent loading, the soft-constrained Mod-PINN shows reduced accuracy because it must learn multiple competing objectives simultaneously. The Mod-HC-PINN mitigates this issue by embedding the cathode boundary and initial conditions into the output structure, thereby reducing the optimization burden and improving physical consistency. The Mod-HC-PINN achieves MAE values of 0.43, 0.41, and 0.27 kPa for the exponential vacuum, ramp surcharge, and cyclic surcharge cases, respectively. Sensitivity analyses further demonstrate that the proposed framework remains robust across practical ranges of network architecture, collocation density, and permeability contrast.