Partial Differential Equations

Seismic wavefield simulation is fundamental to seismology, but conventional finite-difference (FD) methods remain limited by numerical dispersion and stability constraints, which often require dense spatial grids and small time steps and thereby severely limit the effectiveness of iterative inversion workflows. We introduce a conditional diffusion-based wavefield propagator that advances seismic wavefields recursively from one time step to the next. Instead of learning an unconditional data distribution of wavefield evolution, the model is conditioned by a short history of recent wavefield time steps (snapshots), the velocity model, and the wavefield time step index, allowing it to represent the conditional transition between adjacent physical states. By training the network to directly predict the clean next wavefield snapshot, this strong physical conditioning makes it possible to replace the iterative reverse diffusion process with a single network evaluation for each predicted snapshot. To improve stability over long recursive rollouts, we further introduce a causal time-weighted loss, in which adaptive weights, accumulated as exponential moving averages of per-snapshot training errors, emphasize training directions that are consistent with the forward propagation sequence and reduce the amplification of one-step prediction errors. Because the learned propagator is tied to the temporal spacing of the training snapshots rather than to the FD stability limit, it can advance the wavefield using a physical time step ten times larger than that required by the underlying solver. Experiments on the Overthrust, SEG/EAGE, and Marmousi models show that the proposed method accurately reproduces wavefield snapshots and shot gathers and achieves an end-to-end speedup of 2.17 x over a GPU-accelerated tenth-order staggered-grid FD implementation under matched hardware conditions.

Repeated prediction of acoustic fields from spatially distributed boundary excitation is computationally expensive when each source realization requires a new wave simulation. This work introduces a quadrature-aware complex-linear boundary operator (CLBO) that maps complex normal velocity on a vibrating surface to complex pressure at receiver locations. The model couples learned source and receiver basis functions through an explicit complex surface-quadrature contraction, so the boundary excitation enters linearly by construction. This preserves complex superposition, homogeneity, and zero response to zero excitation, while representing the source through coordinates, normals, and quadrature weights rather than a fixed flattened input vector. Reference data were generated using a verified three-dimensional multiple-relaxation-time (MRT) lattice Boltzmann solver and stored in a solver-agnostic boundary-to-field format. CLBO was compared with a fixed-sensor complex DeepONet under matched case splits and optimization settings, with additional tests of structural consistency, receiver-coordinate interpolation, source discretization, source-family holdout, label efficiency, physics-informed ablations, unseen source mixtures, and computational cost. Across five training seeds, CLBO achieved a mean complex relative field error of 0.184 +/- 0.00771, compared with 0.367 +/- 0.00742 for DeepONet. Its measured source-superposition error was 1.31 x 10^-7, and its mean error on newly simulated mixed-source cases was 0.237, compared with 0.415 for DeepONet. Inference was 1.83 x 10^4 faster than the reference calculation for the reported query size. These results show that enforcing the known complex-linear boundary-to-field structure improves physical consistency and generalization under distributed acoustic excitation.

We present CoFINN (Conservation Flux Informed Neural Networks), a physics-informed deep learning framework for predicting compressible flow fields governed by conservation laws. Unlike conventional data-driven convolutional neural networks (CNNs), which optimize only pixel-wise similarity metrics, CoFINN embeds finite-volume conservation physics directly into the training process. Unlike classical physics-informed methods which enforce differential-equation residuals at collocation points through automatic differentiation, CoFINN adopts a finite-volume perspective consistent with modern CFD methodology. CoFINN interprets CNN output fields as structured computational grids, where each pixel represents a finite-volume cell, and enforces conservation consistency through sophisticated numerical flux calculations. The framework is evaluated on transonic flow prediction around airfoils at (M=0.7, Re=6 * 10^6), including challenging conditions involving shock waves and high angles of attack. Results show that CoFINN improves aerodynamic force prediction accuracy, reducing drag prediction error by up to 34% at extreme angles of attack and by approximately 15% on average across the test set. Improvements are particularly significant in limited-data regimes, demonstrating that the conservation-based loss acts as an effective physical regularizer. The proposed approach maintains the computational efficiency advantages of CNN surrogates while significantly improving physical consistency and conservation behavior. The framework is architecture-agnostic and extensible to broader classes of conservation-law-governed physical systems.

Diffusion and Gaussian-interpolant flow-matching samplers approach data through a terminal noise floor $\varepsilon$, a singular limit for manifold-supported or rank-deficient data. We study two properties of a complete sampler specification, comprising its update rule, time grid, and terminal rule. Asymptotic preservation (AP) means a stable and consistent zero-noise discretization with a step count bounded independently of $\varepsilon$. Uniform accuracy (UA) of order $p$ means that, at numerical resolution $h$, the endpoint $W_2$ error is $O(h^p)$ with a floor-independent constant. Bounded log-noise stepping fails AP because its step count diverges. Stopping a stable base solver at a positive switching scale $a$ and appending one map fitted to the analytic normal mode restores AP. On smooth compact boundaryless manifolds, the standard map has exact-input error $O(a^2-\varepsilon^2)$ and sharp zero-floor error $Θ(a^2)$. A base solver with a floor-uniform order-$p$ estimate on the resolved interval retains that order when $a=O(h^{p/2})$, provided the terminal transfer factor remains bounded. Along exact trajectories, the posterior-mean identity $D(x(σ),σ)=x(σ)-σx'(σ)$ cancels the linear terminal defect and enables higher-order fitted maps. A three-evaluation Hermite construction is uniformly third order for exact switching-scale input over $0\le\varepsilon\le a$, and a seven-evaluation construction is fourth order at zero. We classify representative diffusion and flow-matching specifications by AP and UA. On EDM and Rectified Flow checkpoints, a paired decomposition separates base-integration from terminal-completion error and predicts held-out same-seed endpoint errors.

Pulsar timing arrays (PTAs) provide a unique window into nanohertz gravitational waves (GWs), but extracting astrophysical parameters from noisy, long-baseline timing residuals remains computationally challenging with traditional Bayesian techniques due to the high dimensionality of the parameter space, complex and correlated noise models, and the cost of repeated likelihood evaluations. We introduce a Transformer with a physics-informed positional-encoding framework for the efficient inference of eccentric binary black holes in relativistic orbits from PTA data. Our approach embeds analytical GW phase evolution directly into the model through structured positional encodings, enabling the network to learn physically meaningful representations from raw PTA timing residuals. We then use generative models, including discrete and continuous conditional normalizing flows, to infer posterior distributions within a simulation-based inference framework. Across a range of signal-to-noise ratios, the proposed method achieves improved accuracy, sharper posteriors, and faster inference compared to physics-agnostic baselines. While presented for deterministic white-noise signals, the modular framework readily generalizes to realistic PTA analyses incorporating red noise and additional components. This work highlights the potential of physics-aware deep learning models as scalable alternatives to conventional inference pipelines for next-generation PTA datasets.

Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy. In PIEFS, scalar coordinate maps are trained under empirical Gram orthogonality, a supervised linear readout, and a Dirichlet penalty in which the input gradient is transformed by a learnable metric $A(x)=Λ(x)U(x)$. The diagonal factor $Λ(x)$ controls anisotropic scaling, while the orthogonal factor $U(x)$ is parameterized by a structured product of Givens rotations. This construction yields task-adaptive Dirichlet-regularized coordinates rather than eigenfunctions of a fixed supervision-independent operator. Experiments on synthetic, tabular, and image-based benchmarks study the effect of identity, diagonal, and rotation-scaling metrics, and compare the resulting coordinates with classical baselines and NeuralEF. The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.

Convection-dominated convection-diffusion problems often develop thin layers, where the solution has sharp transition profiles and its derivatives are highly localized. This creates a structural mismatch for standard physics-informed neural networks (PINNs), whose trial spaces are not designed to match the value--derivative structure of such layers. We propose a Layer-Resolving XNet Physics-Informed Neural Network (LRX-PINN) based on integrated Cauchy activations. The proposed basis is transition-type at the solution level, while its derivative recovers a localized Cauchy kernel. We show that this structure matches the scaling of convection-dominated layers, inherits the Cauchy approximation mechanism at the derivative-profile level, and identifies $d/\|w\|$ as the effective physical width of a ridge neuron. For analytic layer profiles, this yields derivative-stable exponential approximation in the stretched coordinate and a layer-scaled estimate for the strong residual of the singularly perturbed operator. Numerical experiments on several convection-dominated benchmarks show that LRX-PINN achieves higher accuracy than PIKAN and Fourier-feature PINNs while using less than $30\%$ of their trainable parameters. On more challenging benchmarks, embedding the proposed representation into hp-VPINN-based frameworks further improves the best results obtained by existing hp-VPINN-based baselines without changing their original loss functionals or stabilization strategies. These results show that neural representations aligned with layer structure provide a compact and effective approach for convection-dominated problems.

High-fidelity simulations of complex physical systems are often too expensive for repeated prediction, design, and control. Reduced-order models address this computational cost by compressing high-dimensional dynamics into low-dimensional representations that can be evaluated rapidly, but they often lose accuracy when online dynamics drift beyond the offline training data. Adaptive methods address this limitation by updating the reduced subspace online using external, out-of-span information, such as full-order corrections or sensor snapshots. We discovered that a complementary and previously unexploited in-span adaptation channel exists within the current reduced subspace. To exploit this channel, we continually update the reduced representation using the model's own predictions through an incremental singular-value decomposition with a forgetting factor. This produces a trajectory-informed spectral preconditioner in which the reduced subspace remains unchanged, while the basis is reweighted and realigned according to the directions visited by the evolving dynamics. This internal reorganization prepares the basis to absorb future out-of-span corrections more effectively, improving adaptation without requiring additional external information. We expose the mechanism through a three-dimensional spiral example and demonstrate its benefits on nonlinear partial differential equations, including viscous Burgers and Fisher--KPP dynamics. We also discuss how in-span learning can be interpreted as a dynamical-systems analogue of in-context learning. More broadly, in-span learning suggests a new principle for computational science, revealing that model-generated trajectories contain more usable information than previously recognized.

Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers. The obstacle is increasingly understood to be one of optimisation, owing to the severely ill-conditioned loss landscape. We present $\textbf{DSGNAR}$: Doubly-Sketched Gauss-Newton with Adaptive Ratio, a scalable second-order optimisation framework that confronts this ill-conditioning and, in doing so, obtains unprecedented accuracy and speed. $\textbf{DSGNAR}$ couples a doubly-sketched Gauss-Newton model with a novel strategy that carefully controls both regularisation and step length. Across a suite of problems spanning nonlinear, chaotic, multi-scale, high-dimensional, and Navier-Stokes, the framework greatly improves on the state of the art: able to attain relative $\ell_2$ errors as low as $3\times10^{-16}$ in double precision, improve contemporary results by five orders of magnitude on the canonical Burgers' equation, and as much as eight orders on a high-dimensional Poisson problem, while remaining markedly faster. We further show that, in single precision, solutions at the limit of round-off error can be obtained very quickly: Burgers' equation to $\ell_2^{\text{rel}} = 4.75 \times 10^{-7}$ in under ten seconds. The framework is also robust to the choice of architecture, arithmetic precision, and initial hyperparameters. The code is available at https://www.github.com/wephy/physics-informed-neural-networks

Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics. In this work, we propose a paradigm shift by replacing the discrete training problem of shallow neural networks with a well-posed continuum variational surrogate. We identify a family of $λ$-convex functionals over parameter densities in weighted Sobolev spaces and prove that these variational problems are globally well-posed, stable, and exhibit unexpected almost $C^3$ regularity. Unlike existing Wasserstein-based or Mean-Field approaches, which often face limited regularity and discretization challenges, our formulation provides direct access to elliptic regularity and convex analysis. This allows us to prove that the optimal parameter density can be obtained by solving a single linear system, bypassing iterative optimization entirely. We establish explicit generalization error controls at a rate of $1/α$ relative to the regularization parameter, and prove that finite-width networks of size $N$ achieve the continuum optimum at an $O(1/N)$ rate. This perspective bridges the gap between the Neural Tangent Kernel (NTK) and feature-learning regimes, providing a principled framework for understanding over-parameterization through the lens of variational calculus.

Radio frequency (RF) maps provide a compact representation of multipath propagation characteristics and are fundamental to channel modeling, coverage analysis, and environment-aware wireless optimization. This paper proposes a unified RF map construction framework based on a physics-informed neural network (PINN) and a graph neural network (GNN), supporting both cross-scene generation and in-scene completion with 2D and 2.5D environmental representations. The PINN embeds electromagnetic propagation constraints to establish a physically consistent mapping from receiver locations to multipath parameters, including path gain, time of arrival, and angles, while the GNN enforces spatial consistency by modeling correlations among neighboring receivers. To comprehensively evaluate multipath reconstruction quality, we propose a peak-weighted dynamic time warping metric that jointly accounts for amplitude errors and peak delay misalignment in channel impulse responses. Extensive experiments demonstrate that the proposed method consistently outperforms image-based, diffusion-based, and interpolation baselines across both map-level and multipath-level metrics, achieving robust generalization and high-fidelity RF map construction under sparse observations.

Solving partial differential equations (PDEs) with high-frequency solutions remains a central challenge in physics-informed machine learning due to spectral bias -- the tendency of neural networks to learn low-frequency components preferentially. This paper proposes a Frequency Shift Physics-Informed Extreme Learning Machine (FS-PIELM) framework that addresses this limitation through an additive mechanism for weight initialization. Rather than multiplying random weights by a scaling factor, the method translates the mean of the Gaussian weight distribution while keeping the variance fixed at unity, thereby avoiding the variance amplification inherent in scaling-based methods. Two variants are developed: FS-PIELM-L assigns independent frequency magnitudes to individual neurons, while FS-PIELM-G groups neurons for improved robustness. Theoretical analysis shows that the frequency variance under the proposed framework remains bounded and approaches unity regardless of target frequency, in contrast to the quadratic growth of conventional approaches. The method preserves the computational efficiency of extreme learning machines, requiring only a single linear solve. Experiments on seven benchmark problems spanning six equation types -- Helmholtz, wave, Poisson, Klein-Gordon, heat, and advection-diffusion -- on both regular and complex geometries show that the linear variant achieves the best accuracy in six of seven cases, with improvements of one to nearly five orders of magnitude over existing PIELM variants. The code and data accompanying this manuscript will be made publicly available at https://github.com/xgxgnpu/Physics-informed-vibe-coding/tree/main/FS-PIELM.

Day-ahead wind power forecasting is essential for cost-effective power-system operation. It is primarily driven by future meteorological conditions while retaining temporal dependencies in power generation. In practice, observed wind-farm power often entangles physically available power with local environmental effects and latent operational states, such as shutdowns and curtailment. Existing physical models provide useful constraints but adapt poorly across wind farms, whereas data-driven models can capture rich correlations but often conflate meteorological effects with state-induced deviations. In this study, we propose UniWind, a wind power forecasting model based on physics-informed state routing. UniWind first employs a Physical Prior Estimator to construct a site-calibrated physical prior by combining site-conditioned monotonic warping with a shared physical power curve. It further applies a physical upper-bound constraint to shape this prior as a soft envelope of available wind power generation. UniWind then proposes a Latent State Encoder to model operating-state embeddings and transforms the physical prior into final power forecasts through a State-aware Power Corrector, which uses knowledge-guided supervised state routing and bounded, state-specific expert correction. Full-shot and cross-farm zero-shot experiments on more than 20 real-world datasets demonstrate the accuracy and robustness of UniWind.

Predicting complex spatiotemporal dynamics in physical processes often demands computationally expensive numerical methods or data-driven neural networks that suffer from high training costs, error accumulation, and limited generalizability to unseen parameters. An effective approach to address these challenges is leveraging physics priors in training neural networks, known as physics-informed deep learning (PiDL). In this work, we introduce the Multi-Resolution Finite-Volume-inspired network, MuRFiV, designed to capitalize on the conservative property of finite volume on the global scale and the expressive power of deep learning on the local scale. We demonstrate the effectiveness of MuRFiV on several spatio-temporal systems governed by partial differential equations (PDEs), including Burgers' equation, shallow water equations, and incompressible Navier-Stokes equations. By embedding PDE information into the deep learning architecture, MuRFiV achieves strong long-term prediction accuracy and remains stable over very long autoregressive rollouts, significantly outperforming data-driven neural network baselines. This result highlights the promise of combining multiresolution learning with finite-volume-inspired inductive bias for accurate and robust long-term prediction of complex dynamics.

Generative models have emerged as scalable surrogates for physical simulation, yet they offer no guarantee that their outputs respect the conservation laws, boundary conditions, and nonlinear invariants that govern the underlying physics. Constrained sampling closes this gap, enforcing such constraints exactly at inference time without retraining, but at a computational cost: projection, correction and trajectory-optimization steps are repeated during sampling, with these steps becoming expensive for nonlinear constraints. Standard ML frameworks exacerbate this: their dense tensor algebra and limited sparse solver composability obscure the structure that physical constraints naturally induce, making efficient batched nonlinear optimization difficult to realize in practice. We address this bottleneck by exploiting the structure that sample-wise batching and local PDE couplings induce in the projection subproblems -- namely, block-sparse Jacobian and KKT systems -- exposing this structure using ExaModels.jl and solving the resulting sparse nonlinear programs with MadNLP.jl and GPU sparse factorization. Applied to Physics-Constrained Flow Matching (PCFM), on PDE benchmarks with linear, nonlinear, one-dimensional, and two-dimensional constraints, this approach accelerates nonlinear constraint projection while maintaining constraint satisfaction. These results show that sparse GPU nonlinear optimization is a practical foundation for constrained generative sampling in scientific machine learning.

Accurate modeling of the divertor temperature field is essential for preventing material melting and damage and for extending the service life of fusion devices. However, conventional numerical methods, such as the Finite Element Method (FEM), are computationally expensive and therefore unsuitable for real-time applications. Therefore, a fast and generalizable method is required for real-time reconstruction of the divertor temperature field and subsequent real-time control. To address the above issue, we propose a Physics-aware Neural Operator Transformer (PNOT) to characterize the spatiotemporal evolution of the divertor temperature field. It models boundary heat-flux relations as a structured graph and employs graph attention to explicitly capture spatial physical dependencies. Inspired by physics-aware attention, we further develop a physics-aware neural operator module to aggregate query points with similar physical conditions via slicing and model heat diffusion, while a gradient-constrained Sobolev regularization loss enforces consistency between function values and their derivatives. Experimental results show that these physical constraints improve prediction accuracy while preserving physical consistency. The source code of this paper will be released on https://github.com/Event-AHU/OpenFusion

Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error. Truncation-based methods often require problem-dependent artificial boundary conditions, while global spectral bases may be inefficient for localized structures, irregular geometries, or solutions with different near-field and far-field behaviors. We propose a domain-decomposed randomized neural network framework for such problems. Different randomized subnetworks are assigned to different spatial regimes: a near-field subnetwork captures local and geometric features, whereas a far-field subnetwork represents exterior decay. The subnetworks are coupled by boundary and interface conditions, and only the output-layer coefficients are solved from linear least-squares systems arising from Petrov--Galerkin or collocation formulations. We develop a Petrov--Galerkin method for semi-unbounded elliptic problems and a collocation method for fully unbounded, perforated, and time-dependent problems. A conditional bounded-parameter approximation result is proved in a broken Sobolev norm, together with an error decomposition covering approximation, empirical-consistency/quadrature, and least-squares optimization errors. Numerical experiments for Poisson and time-dependent Schrödinger equations demonstrate the accuracy and flexibility of the proposed method.

We present LINet (Linear Integration Network), a Multi-Stream Neural Network (MSNN) for RGB-D scene classification. Current multi-modal architectures treat feature fusion as a discrete, ad-hoc event: early fusion entangles representations prematurely, late fusion isolates them until the final layer, and hybrid or attention-based methods require architectural guesswork to place intermediate fusion blocks. LINet addresses this structural compromise by maintaining three dedicated parallel streams (RGB, depth, and integration) where a novel Linear Integration Convolution (LIConv2d) operator enables continuous cross-modal learning at every layer. The integration stream receives raw filtered signals from both modality streams and combines them before the nonlinear activation threshold, conceptually inspired by somatic integration preceding the neuronal firing decision. Implementing continuous integration exposes a critical initialization pathology: Kaiming initialization of the bridging weights scrambles gradients before they reach the stream backbones, producing a failure mode that resembles overfitting but is corrupted gradient flow. A 1/N constant initialization mitigates this. We employ progressive modality dropout, a curriculum adapted to continuous fusion in which blanking probability increases from zero, preventing pathway collapse, a form of negative co-learning, by forcing robust independent stream representations. Trained from scratch on SUN RGB-D 19-class scene classification, LINet reaches 45.2% mean class accuracy at ResNet18 scale, outperforming prior from-scratch results, and rises to 49.6% with in-domain RGB-D (ScanNet) pretraining.

Generative Astrodynamics is advanced in this work by extending generative modelling to an orbit determination problem in the cislunar environment. The task is formulated as conditional density estimation, aiming to infer the probability distribution of the initial state from angles-only measurements over short observation arcs. A normalising flow is trained on perturbed topocentric observations from Near Rectilinear Halo Orbits, enabling a flexible and potentially multimodal posterior representation. Given new measurements, the learned density is sampled to generate statistically consistent and physics-informed state hypotheses. These estimates are refined via nonlinear least-squares minimisation, providing a competitive warm start for classical algorithms.

Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coarse-grid correction can discard the local phase and propagation-direction information carried by oscillatory errors. We propose Multi-channel Multigrid (McMg), a learned phase-space multigrid preconditioner for heterogeneous Helmholtz equations. Rather than predicting the solution directly, McMg maps residuals to corrections within an iterative framework. Its central idea is to coarsen physical space while retaining unresolved local wave information in the channel dimension: each coarse node carries a learned packet of amplitude, phase, direction, and scattering coefficients rather than a single scalar unknown. The architecture combines linear multi-channel transfer operators with locally adaptive stencils, neural PDE operators, and medium-dependent smoothers whose coefficients are generated from the wave speed. For a fixed medium, the V-cycle is linear in the residual; nonlinear physical features are computed once in a setup phase and cached, so each online iteration reduces to convolutions with fixed coefficients. We further study generalization across scales. Models trained on small domains transfer directly to larger domains and higher effective wavenumbers, and a Layer-by-Layer Progressive Finetuning (LLPF) strategy improves large-domain scalability by adding new coarse levels while finetuning only the newly introduced parameters. Numerical experiments on high-frequency, high-contrast, and large-scale three-dimensional problems demonstrate that McMg requires substantially fewer iterations and less wall-clock time than strong classical baselines, while consistently outperforming existing neural preconditioners.

Discovering governing equations directly from observational data is a key step towards interpretable scientific machine learning. Current data-driven approaches typically operate on a single dataset, inherently limiting their performance when faced with restricted observations. In practice, multiple datasets are often available for the same physical system, distinguished only by distinct initial conditions or boundary configurations. Here, we present a competitive optimization framework designed to discover shared partial differential equations (PDEs) from multi-source datasets, termed MCO-PDE. The framework first trains independent neural surrogates for each data source, and then employs a soft-competitive weighting mechanism to dynamically assess dataset credibility and aggregate a consensus global coefficient. Integrated with a genetic algorithm for structural search, this approach simultaneously identifies the functional forms and parameters of the governing laws. We demonstrate that fusing as few as 50 observations per dataset across seven cases recovers canonical equations with high accuracy. The framework inherently handles two- and three-dimensional domains characterized by irregular boundaries and heterogeneous coefficients, and successfully extracts physically meaningful laws from real-world wave-tank experiments. Overall, this work establishes a promising route for automated scientific discovery via heterogeneous data fusion.

Data-driven material modeling techniques have gained significant attention due to their ability to capture complex constitutive behaviors beyond the limitations of classical material models. Physics-augmented neural networks (PANNs), which embed physical constraints directly into their architecture, combine the flexibility of machine learning with the reliability required for engineering simulations. This work presents an approach to integrate such network architectures into the explicit finite element solvers Simcenter Radioss and OpenRadioss (Siemens). A framework for transferring pretrained network architectures and their parameters to a standalone user material routine is developed. Networks are trained using PyTorch, though the procedure can be adapted to other frameworks such as TensorFlow, enabling the use of PANNs within existing finite element technology without requiring specialized solvers. Particular emphasis is placed on computational efficiency. The influence of network architecture on simulation performance is investigated, and strategies for reducing evaluation costs while preserving accuracy are discussed. Specifically, replacing the SoftPlus activation function with SQuarePlus is shown to reduce computational cost. A publicly available GitHub repository automates the generation of Fortran user material routines, requiring only the specification of the network architecture and trained parameters. An example impact simulation demonstrates that the generated PANN user material reproduces the nonlinear behavior characteristic of hyperelastic materials under large strains, providing a practical route toward machine-learning-based constitutive models in explicit finite element simulations.

Calibration remains one of the principal obstacles to the deployment of machine learning in scientific instrumentation because it typically relies on expert intervention, dedicated procedures, and manually labelled data. We introduce a physics-informed self-supervised framework that jointly learns latent detector calibration parameters and task-specific predictions directly from raw measurements without requiring pre-calibrated signals or external labels. The method exploits known physical constraints to generate pseudo-labels iteratively, transforming calibration into a self-supervised optimization problem. The approach is demonstrated for ionic charge-state determination in the VAMOS++ magnetic spectrometer, where the calibration of a segmented ionization chamber and the inference of ionic charge states are learned simultaneously. Starting from a weak prior on the mean ionic charge state, the model progressively refines its predictions through iterative fractional pseudo-labelling driven by the discrete nature of atomic masses. Beyond accurate ionic charge-state reconstruction, the inferred calibration coefficients provide a compact representation of the detector state that enables automated monitoring of gain drifts, pressure variations, and detector aging. The resulting labels can subsequently be transferred to specialized models that quantify detector imperfections and track their spatial and temporal evolution. These results establish a general paradigm for self-calibrating and self-monitoring scientific instruments and represent a step toward intelligent experimental systems capable of autonomous calibration, analysis, and performance optimization.

Diffusion and continuous-flow generative models achieve high-quality generation, and their deterministic sampling can be formulated as solving learned ODE dynamics. However, accurate ODE discretization often requires many steps, making efficient few-step generation a key challenge. Among acceleration strategies, reflow-based distillation simplifies teacher ODE trajectories so that a student model can approximate the teacher transport with fewer steps. We identify a theoretical limitation of this paradigm, namely that trajectory matching can under-determine the distribution induced by the student model. In particular, two student models can attain the same trajectory-matching loss while inducing different endpoint marginal distributions, which may lead to different generation quality. To address this limitation, we introduce a marginal-alignment regularizer that penalizes the discrepancy between the student-induced marginal and the corresponding teacher marginal at the endpoint of each distillation interval. The regularizer is computed by tracking log-density changes along the ODE induced by the student model and evaluating scores from the frozen teacher model, without requiring auxiliary trainable networks or adversarial optimization. The resulting framework applies uniformly to the reflow family, including vanilla reflow and piecewise reflow. We further prove a telescoping total-variation bound showing that local marginal alignment controls the final-time discrepancy between the student-induced and teacher-induced distributions. Experiments on benchmark backbones demonstrate the effectiveness of the proposed method for few-step generation.

Quantum kernel estimation on near-term hardware is shot-budgeted: every entry of the kernel Gram matrix is a Bernoulli expectation that must be sampled with a finite number of circuit executions. Recent work on quantum kernel classification has shown that allocating shots non-uniformly across kernel entries, weighted by their downstream task sensitivity, can reduce the shot budget required to reach a target accuracy. We extend this idea to Gaussian process (GP) regression, a setting whose downstream quantities (full-spectrum posterior variance, log-determinant, marginal likelihood) couple to kernel error more tightly than the sign-only outputs of classification. We derive three closed-form pair-level sensitivities predictive coupling $|α_iα_j|$, leave-one-out residual, and marginal-likelihood gradient and plug them into a Neyman-style minimum-variance allocation rule. To prevent catastrophic over-concentration when the warm-up sensitivity estimate is itself noisy, we add a high uniform coverage floor justified by a Frobenius lower bound on the missing-entry perturbation. On four UCI benchmarks and two synthetic RBF + Bernoulli controlled studies, the resulting allocator delivers $10$--$21\%$ test-RMSE improvement over uniform allocation across the moderate-budget regime. The gain transfers (i) to genuine ZZ and Pauli-Z quantum kernels on quantum-natural data ($-13$--$15\%$ at low budget, $p<0.05$ paired) and (ii) to four downstream tasks (Bayesian quadrature, heteroscedastic regression, hyperparameter learning, multi-output Cokriging). On UCI features embedded into a ZZ kernel the gain disappears, consistent with the exponential-concentration regime where shot allocation has nothing to exploit.

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving nonlinear partial differential equations (PDEs), including battery electrochemical models. They typically en-force conservation laws within the loss function to ensure physically consistent solutions. Tradi-tional numerical methods such as finite difference, finite volume, and finite element techniques, re-ly on discretization and can be computationally expensive for nonlinear systems. To address this challenge, PINNs offer improved scalability, particularly for reduced-order models like the single particle model with electrolyte (SPMe). The SPMe describes lithium-ion battery dynamics through coupled diffusion, transport, reaction kinetics, and voltage equations. Despite these advantages, training SPMe-based PINNs from scratch for different battery chemistries or operating conditions is demanding and often leads to slow convergence. To overcome this limitation, this work introduces a transfer learning framework for SPMe-PINNs. The model is first pretrained to learn general elec-trochemical dynamics and then adapted to a target battery by transferring weights, freezing se-lected layers, and fine tuning the remaining parameters, including estimating key electrochemical variables. Validation using PyBaMM demonstrates accurate voltage prediction, indicating that the proposed approach preserves electrochemical consistency while reducing training time and ena-bling efficient generalization across batteries.

Physics-informed neural networks (PINNs) have recently emerged as a promising framework for addressing the Calderón inverse problem from limited boundary data. In this work, we revisit neural Calderón inversion by introducing multiscale boundary excitations based on randomized wavelet functions and investigating the role of Fourier-feature encoding (FFE) for representing sharp conductivity variations. We propose a physics-informed reconstruction framework that represents the unknown conductivity and the associated family of electric potentials with separate neural networks conditioned on the applied boundary excitations. The governing elliptic PDE is enforced through physics-informed residuals, while finite Dirichlet-to-Neumann (DtN) data are incorporated through boundary losses. Using synthetic data from a finite-difference forward solver, we evaluate the method on conductivity fields with inclusions, sharp interfaces, smooth profiles, and heterogeneous media. Results show that the framework recovers dominant conductivity structures from finite boundary measurements with relative errors between $3\%-12\%$ approximately. We show that FFE improves the reconstruction of localized sharp features, particularly for inclusions and interfaces, but are not universally optimal, with raw-coordinate networks performing competitively for smoother fields. These results highlight coordinate representations and boundary excitation design as key factors in neural Calderón inversion.

We introduce a physics-constrained neural network for the rapid prediction of rigorous coupled-wave analysis outputs in the form of Jones matrices. Starting from energy conservation in lossless layered periodic structures, we use the fact that the scattering outputs lie on a Stiefel manifold. This energy constraint is enforced as a hard condition by projecting onto the manifold using differentiable symmetric orthogonalization. The resulting surrogate enforces energy conservation by construction while preserving differentiability for gradient-based inverse design. The performance and generality of the proposed approach are demonstrated through the inverse design of a diffractive waveguide combiner for augmented reality glasses.

Differentiable partial differential equation (PDE) solvers underpin solver-in-the-loop ML training, gradient-based optimal control, and inverse problems, yet the practical cost of obtaining correct, usable gradients from a given solver on a given problem is largely undocumented. Integration effort, computational cost, gradient accuracy, and numerical conditioning vary widely across solvers and are discoverable only by trial and error. We introduce Mosaic, an extensible benchmarking framework for differentiable PDE solvers that standardizes access to solver gradients. Each solver is packaged as a containerized component (Tesseract) exposing a uniform gradient API regardless of language or automatic differentiation (AD) strategy, enabling researchers to evaluate, compare, and build on non-trivial physical solvers. Our evaluation of 14 solvers across fluid dynamics, structural mechanics, and heat transfer demonstrates that the benchmark surfaces practically relevant differences: order-of-magnitude variation in computational cost and Jacobian conditioning, alongside structural incompatibilities that eliminate solvers from realistic tasks entirely. Despite this variation, all solvers that produce gradients converge to similar optima, indicating that the practical barriers are memory limits, numerical stability, and setup compatibility rather than gradient accuracy alone. Mosaic is open-source and available at https://github.com/pasteurlabs/mosaic.

Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution. Recent hybrid methods promote physical correctness by targeting the PDE residual via gradient descent or Gauss--Newton steps, but inherit the compute cost and instability of the underlying classical optimizers. We show, theoretically and empirically, that numerically minimizing the PDE residual can be an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining why these methods often do not make accurate predictions despite achieving low residuals. We propose error-conditioned Neural Solvers (ENS), built on a different principle: rather than an optimization target, the PDE residual field is passed as a direct input to the network at each iteration, enabling it to read the spatial structure of its own errors and learn an update policy to iteratively correct its predictions. Across four PDE families, ENS attains the highest prediction accuracy in the large majority of settings, with gains reaching $10\times$ on turbulent Kolmogorov flow, while avoiding the expensive compute cost of hybrid methods. ENS's learned correction policy generalizes under distribution shift, including zero-shot parameter changes and cross-equation transfer, where its relative advantage is largest in the ill-conditioned regimes where residual minimization is least reliable. Project website: https://neuralsolver.github.io/.

We develop the fTNN, a deterministic tensor neural network subspace method for problems involving the fractional Laplacian on bounded domains, taking the fractional Poisson equation and time-dependent fractional advection-diffusion equation as typical representatives. The work employs a geometry-adapted integration split featuring a spatially dependent near-field radius, which decomposes the fractional Laplacian into three contributions: a singular near field, a regular interior far field, and an analytical exterior far field. Then the singular radial integrals are treated by Gauss-Jacobi quadrature, the regular radial integrals by Gauss quadrature, and the angular variables by deterministic angular quadrature, yielding a fully deterministic integration framework of the fractional Laplacian operator. To accurately resolve low-regularity solutions and the associated loss functional, we construct boundary-singularity-aware trial functions enriched with explicit boundary features, and propose two strategies for automatically selecting the leading exponent and evaluating the loss function from the singularity structure induced by the fractional operator, or jointly by the fractional operator and the source term. For time-dependent fractional PDEs, we design a spatiotemporally separable neural network that factorizes the time-space residual into a sum of low-dimensional temporal and spatial integrals, and we integrate this representation with an alternating neural network subspace optimization strategy for efficient training. Numerical experiments show that the proposed framework attains high accuracy on the tested benchmarks and improves substantially over existing fPINN and Monte Carlo baselines, particularly for problems with strong boundary singularities and long-time simulations.

Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency. Identifying the system Hamiltonian from noisy observations of state variables is a challenging task. For simulations to faithfully reflect the long-term behavior of Hamiltonian systems, especially energy conservation, it is essential to use symplectic integrators, which preserve the system's geometric structure. This fidelity comes at a cost: implicit symplectic integrators are more computationally intensive and make backpropagation through the ODE solver non-trivial. However, by leveraging the fact that symplectic discretizations of the adjoint system yield the same sensitivities associated by backpropagation, we obtain an efficient method of training the Neural Network parameters. In our work, we explore this alternate method of HNN training under noisy observation of trajectories with our HNN model based on an implicit symplectic integrator. Computationally, a predictor-corrector based ODE solver and fixed point iteration help to mitigate the computational cost of the implicit timestepping, resulting in more efficient generation of gradient updates. We showcase the numerical advantage, in experiments, in system identification and energy preservation on a range of non-separable, chaotic systems and the efficient computation and memory complexity of our method. We also observe that the post-processing of the learned Hamiltonian using backward error analysis yields a modified Hamiltonian that is a more accurate approximation of the true Hamiltonian without the need to use more accurate discretizations of the flow map.

We introduce an accelerated Langevin-based sampling method that is based on two complementary devices: \emph{SamAdams} adaptive timestepping, which automatically shrinks the effective integration step in stiff regions of phase space using a relaxed stiffness monitor, and \emph{position-adaptive Langevin} (PAL) dynamics, which concentrates friction along the local force direction while preserving the canonical distribution as the exact invariant measure. The resulting combined scheme (SA-PAL) is implemented in a palindromic integrator which requires only one force evaluation per iteration through suitable organisation of the integration steps and by exploiting the rank-one-plus-scalar structure of the PAL friction tensor. We test the method on various model problems: the Rosenbrock function, a thin entropic channel, the Mueller-Brown potential, and a Bayesian parameterisation problem with a sparsity-inducing shrinkage prior. On the Rosenbrock and Mueller-Brown potentials mixing rates are improved by 1.5-3 times compared to fixed stepsize integration. Efficiency gains of more than an order of magnitude are documented in the other examples.

As semiconductor technology nodes scale, computational lithography is essential for ensuring yield and performance. However, lithography is a continuous physical process involving mask optimization, optical imaging, resist exposure, and development, which existing models fail to capture. To overcome this limitation, we present LithoDreamer, the first physics-informed World Model (WM) framework for computational lithography, which formulates the ``Layout-Mask-Resist Image-After Development Image (ADI)'' pipeline as a decision-driven multi-step evolution system. LithoDreamer captures feature changes between adjacent states to model stage-specific physics-informed latent spaces, in which it controls process intervention exploration and drives subsequent state transitions. To achieve interpretable intervention optimization without continuous supervision, we propose a contrastive variational optimization paradigm that contrasts the latent differences between intervention paths with variational evolution constraints, guiding the model to generate evolutions consistent with real lithography physics. Experiments show LithoDreamer achieves state-of-the-art performance in forward evolution and inverse planning. Our lithography dataset is publicly available at GitHub (https://github.com/7jiangyq/lithodreamer.git).

Physics-informed neural networks (PINNs) embed governing equations in their loss function, enabling mesh-free solutions to partial differential equations. Low training loss is treated as evidence that the learned solution is physically correct. This paper shows that assumption breaks down when encoded physics are incorrect. By perturbing PDE parameters before training, a setting we describe as physics parameter poisoning or parameter misspecification, we produce models that train to low loss but give incorrect answers; we treat the perturbation schedule as sensitivity analysis rather than only as a security threat, and none of our claims requires an adversary. Achieving low residual loss does not discriminate accurate from inaccurate solutions: poisoned models reach losses at or below the clean baseline yet differ by large margins, so driving the residual down is not evidence of physical accuracy. Across three PDE systems (Burgers equation, Navier-Stokes cavity, and convection-diffusion), poisoned models match or beat the clean-model training loss while their solutions differ by up to 71% in the fixed sweep and up to 128% under adversarial search; at Cavity Re=400 the poisoned loss falls below the clean baseline. We define a detection difficulty ratio R (solution error divided by training loss) to summarize how invisible the corruption is, though cross-PDE comparison is complicated by differences in loss scale. We test six candidate defenses, none of which reliably detects corruption across all regimes. We propose a post-hoc defense: sweeping the PDE residual loss across parameter values without retraining. The loss minimum recovers the true training parameter without external data, and generalizes across all three PDE systems. The effect holds across five network architectures (8.7K to 133K parameters), is bidirectional, and is confirmed across multiple random seeds.

High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging. Existing deep learning solvers often rely on repeated automatic differentiation to evaluate differential operators, which can cause instability and amplify derivative errors in high dimensions, while probabilistic methods based on stochastic representations require explicit knowledge of the data-generating dynamics and therefore do not apply to black-box environments. We introduce two types of simulators as data-generating mechanisms, and take a ``representing-then-learning" approach that learns the solutions and their derivatives under settings where the underlying PDE operators are accessible only through simulations and pointwise evaluations. Our representation of derivatives relies on the zeroth-order derivative (ZOD) estimators derived from perturbed Monte Carlo trajectories. This fully model-free approach generates targets for the gradient and Hessian networks using only function evaluations. We provide a statistical learning analysis of the proposed approach, including a bias--variance tradeoff for ZODs. Assuming a standard contraction property of the underlying operator, we establish a non-asymptotic error bound that decomposes the total error into discretization error, approximation error, statistical error, and ZOD bias. Crucially, we derive the sample complexity of the learned representations in (weighted) Sobolev space, characterizing the error up to second-order derivatives. Numerical experiments illustrate the competitive performance of the method in moderate and high dimensions.

Phase-field models play a central role in the continuum description of phase separation, in which the bulk free-energy density and the interfacial thickness parameter determine pattern formation and microstructural evolution. In practice, these constitutive quantities are rarely known a priori and must be inferred from limited dynamical observations. In this work, an extended pseudo-spectral physics-informed neural network (ESPINN) framework is developed for the inverse identification of phase-field models from transient snapshot data. It enables the simultaneous recovery of both the bulk chemical potential and unknown gradient coefficients. Numerical experiments on the one-dimensional Cahn-Hilliard equation demonstrate accurate and statistically stable reconstruction in the noiseless regime, with substantial constitutive information recoverable from even a single snapshot pair. In the presence of noise, reconstruction accuracy degrades gracefully, and increasing the number of snapshots improves robustness by reducing variance across runs. These results establish ESPINN as a data-efficient and physically consistent approach for learning free-energy structure in continuum models of phase separation.

In this paper, we introduce two neural-network-based numerical schemes for solving systems of coupled ergodic Backward Stochastic Differential Equations (eBSDEs), motivated by the approximation of optimal strategies within the framework of forward utilities in a regime-switching stochastic factor model. Our approach builds on the representation of such models through systems of eBSDEs introduced in [HLT20]. We first establish a link between the solution of the system of ergodic BSDEs and that of an associated multidimensional BSDE with random terminal time, given by the hitting time of the positive recurrent stochastic factor. Building on this representation, we introduce a locally additive deep learning scheme obtained by minimizing aggregated local error terms. We then present a new Deep Galerkin Method (DGM) inspired algorithm that minimizes the residual of the associated ergodic PDE system, relying on a representation of the ergodic cost. Finally, we apply this framework to regime-switching forward utilities in a stochastic factor model. We first derive a general consistency SPDE that characterizes regime-switching forward utilities and retrieve their representation with systems of ergodic BSDEs in the homothetic case. Numerical experiments demonstrate the performance of the proposed methods, with a particular focus on the impact on forward preferences of taking into account regime switches.

Wood materials exhibit complex, spatially varying thermal properties that challenge traditional architectural assumptions of material homogeneity. Although data-driven approaches can directly map wood RGB images to their corresponding thermal responses, they operate as uninterpretable black boxes that prioritize statistical correlation and may absorb experimental noise rather than thermodynamic plausibility. To address these limitations, we present physics-informed deep learning frameworks that integrate partial differential equations (PDEs) to predict pixel-level thermal responses of spatially heterogeneous wood materials using wood RGB images and testbed temperature maps. Specifically, we investigate two distinct approaches to enforcing a normalized 2D steady-state heat transfer equation derived from the general heat transfer equation: Physics-Informed Convolutional Neural Networks (PICNNs), which embed physics as a soft penalty term in the loss function, and Physics-Integrated Convolutional Neural Networks (PInteCNNs), which hard-code an analytical approximator-predictor-corrector solver directly into convolutional neural networks. To validate our proposed approaches, we collect three real-world multimodal datasets of Poplar, Grandis Cross-Cut (Grandis-CC), and Grandis Radial-Cut (Grandis-RC) wood samples. We further demonstrate that embedding physical inductive biases successfully balances predictive accuracy, physical interpretability, and intra-species diversity, outperforming data-driven approaches in handling complex wood material heterogeneity and enabling the extraction of interpretable physical parameters. Project: https://zekifayes.github.io/pim

Physics-informed neural networks (PINNs) provide an effective way to solve partial differential equations (PDEs) by embedding physical principles into the learning process. However, the conventional PINN formulation, in which all constraints are imposed as soft penalty terms within a composite loss, often exhibits slow convergence, sensitivity to loss weight scaling, and inaccurate boundary enforcement due to poor conditioning of the optimization landscape. To address these limitations, this study proposes a unified hard--soft physics--informed neural network (HSPINN) with adaptive loss weighting. In this framework, Dirichlet and periodic boundary conditions are enforced exactly by construction through analytical or polynomial lifting, masking functions, and periodic feature mappings, while the governing PDE residuals, Neumann fluxes, and initial conditions are treated as soft constraints. An inverse-share softmax strategy dynamically balances the relative importance of individual loss components during training, eliminating manual penalty tuning and improving gradient stability. This formulation ensures boundary admissibility throughout optimization and enhances convergence efficiency and numerical robustness. Applications to representative elliptic (Poisson), parabolic (Burgers), and hyperbolic (convection with periodic boundaries) problems demonstrate that HSPINN consistently achieves faster convergence, higher accuracy, and greater stability than conventional PINNs, establishing a general and scalable foundation for physics-constrained deep learning across science and technology.

High-fidelity simulations of free-surface flows using Lagrangian methods such as the Particle Finite Element Method (PFEM) are computationally demanding due to continuous domain updates and repeated solution of the governing equations. This challenge is further amplified by non-Newtonian rheologies, where material nonlinearities increase computational cost. These limitations motivate the development of efficient surrogate models to approximate PFEM dynamics at reduced cost. While data-driven deep learning approaches are promising, a key challenge is designing models that operate on arbitrary and evolving geometries. We propose a self-attention-based neural surrogate for PFEM simulations of free-surface flows. The architecture leverages attention mechanisms to model node interactions and capture complex spatial dependencies, while preserving the PFEM mesh discretization. This provides a geometric and topological framework for remeshing and node redistribution, maintaining high-quality spatial discretization during rollouts, improving long-term stability, and enabling reconstruction of derived mechanical quantities via standard finite element operators. Two attention formulations are considered: a standard self-attention mechanism and a linear variant that reduces computational cost and improves scalability. The models are evaluated on two- and three-dimensional free-surface flow benchmarks with evolving geometries, varying material parameters, and non-Newtonian fluids. Results show accurate prediction of transient dynamics and final configurations, with significantly improved scalability. The mesh-based formulation also enables direct reconstruction of quantities such as stress fields. Overall, the framework provides an accurate and scalable surrogate strategy for PFEM simulations in engineering-scale applications.

Jamming and spoofing pose significant threats to wireless and satellite navigation by disrupting radio-frequency (RF) signals and compromising availability and integrity. Robust RF interference direction finding through angle-of-arrival (AoA) estimation is therefore essential for detecting and localizing anomalous signals. Although data-driven methods perform well under line-of-sight (LoS) conditions, their performance degrades in practical environments due to non-line-of-sight (NLoS) multipath propagation. In this work, we propose a hybrid learning framework that incorporates physics-informed constraints into deep neural networks to improve the robustness of AoA estimation. A neural network is trained to estimate the azimuth and elevation of incoming signals received by a four-element antenna array, while a physics-informed loss enforces consistency between the predicted angles and inter-antenna phase differences under a plane-wave model. We further introduce a latent-space classifier to distinguish LoS from NLoS samples. Since inter-antenna phase differences under LoS propagation exhibit domain-invariant structure across environments, the physics-based loss is applied only to LoS samples, promoting physically consistent and domain-invariant representations without over-constraining the model in NLoS scenarios. In addition, domain-incremental learning (DIL) across NLoS environments with varying scatterer distributions improves cross-domain generalization. Evaluations on real-world datasets show that the proposed method reduces AoA estimation error by up to 6° in low-exemplar settings compared with DIL baselines.

Generative models have revolutionized the process of materials discovery, yet they often fail to satisfy underlying physical causality. Through an analysis of Large Language Models (LLMs) augmented with knowledge graphs derived from current literature, we uncover a phenomenon termed contextual tunneling, where models "over-anchor" on narrow, retrieved evidence while suppressing global physical reasoning. To address this problem, we introduce ARIA, a causal-aware framework that conditions knowledge use on mechanistic completeness. ARIA routes each query through a three-tier cascade: (i) direct causal reasoning when complete evidence chains of Process-Structure-Property (PSP) are available, (ii) physics-informed analogical transfer for sparse or novel material systems, and (iii) explicit parametric fallback when external evidence is incomplete. As a proof of concept, we construct a Knowledge Graph (KG) containing 2,839 extracted PSP relations from peer-reviewed articles in the materials literature and evaluate ARIA on forward prediction and inverse design tasks for two-dimensional (2D) materials. ARIA mitigates contextual tunneling, improves over unaugmented and naive KG-augmented baselines, and provides further gains when an online literature search is used for evidence enrichment. Crucially, ARIA produces auditable causal traces, enabling physically grounded and trustworthy AI-assisted materials discovery.

Physics-informed neural operators (PINOs) have recently gained attention as fast numerical simulators with potential for solving inverse problems. This study proposes the first PINO-based method for speech production analysis. The model learns the governing one-dimensional wave equations directly without requiring pre-computed supervised training data. Using vocal tract shape data as input features, we compare the proposed model's predicted f0, glottal volume velocity and sound pressure at the lip for five static vowels to a conventional Runge Kutta/Finite difference approach. With errors of 0.8% for glottal volume flow and 3.2% for speech waveforms, the proposed model enables efficient GPU-parallelized simulation without iterative calculations. We conclude that PINO is a promising approach for fast analysis of speech.

Solving high-dimensional and high-order PDEs is challenged by the coupled growth of spatial dimensionality and derivative order. Recent stochastic derivative estimators reduce this cost by replacing full derivative tensors with randomized dimension or Taylor estimators, but they are mostly designed for fixed physical parameters and require retraining for each new parameter. We show that direct conditional parameterization of such solvers entangles physical parameters with the high-order automatic differentiation graph, causing extra memory growth and parameter-induced variance amplification. We propose Parameterized Representations via Implicit Stochastic Modulation (PRISM), a plug-and-play framework for parameterized high-dimensional and high-order stochastic neural PDE solvers. PRISM uses a hyper-generator to map physical parameters to affine modulators that scale and shift a purely spatial latent manifold, while keeping parameter branches value-connected but spatial-tangent-disconnected. This design preserves unbiased stochastic dimension and Taylor estimators, removes the parameter encoder from high-order spatial AD, and provides a variance-aware Lipschitz envelope over the parameter space. We prove parameterized unbiasedness, estimation-error bounds, and convergence under bounded stochastic variance. Experiments with PRISM-STDE and PRISM-SDGD on nonlinear parameterized PDEs show stable zero-shot generalization, reduced memory usage, and scalability up to 100,000 dimensions on a single GPU, with efficient low-rank SVD adaptation for unseen parameters.

The Morse index of a free boundary minimal surface is encoded in its Jacobi-Steklov spectrum, and we test how faithfully a physics-informed neural network (PINN) reproduces that spectrum on a problem whose answer is already known in closed form. The benchmark is the critical catenoid in the unit ball $\mathbb{B}^3$, where it is well known that the Morse index equals $4$ and the nullity equals $2$. Separating the angular variable reduces the eigenvalue problem to a family of one-dimensional Robin problems on $[-T,T]$, one for each Fourier mode. A network that enforces the parity of each mode by construction, and carries the eigenvalue as a trainable parameter, returns the three eigenvalues below the stability threshold to within $10^{-6}$ to $10^{-4}$ of their exact values, with PDE residuals of order $10^{-4}$; assembling them recovers the index $4$ and the nullity $2$. We then track the spectrum along a one-parameter homotopy joining a flat reference operator to the catenoid Jacobi operator and identify the crossings at which the index changes. Since the critical catenoid is rigid, a fact we prove, this homotopy deforms operators rather than surfaces. We close by explaining how the same pipeline, with its one-dimensional solver replaced by a two-dimensional one, is poised to address genuinely geometric families in ellipsoidal balls, where the boundary curvature is no longer constant, and the Morse index is not yet known.

In this paper, we present a fully discrete approach for the accurate and numerically efficient dynamical modeling and state estimation of continuum robots. The model is based on geometrically exact beams in a minimal, strain-based formulation and derived in the framework of Lie group variational integrators, allowing to preserve important geometric properties that we exploit to achieve high accuracy and numerical efficiency. We then propose a disturbance observer based on an extended Kalman filter formulation that reliably estimates system states as well as model uncertainties and external disturbances. Experiments on a real system validate the accuracy and efficiency of the proposed model and observer.

We introduce a semi-parametric framework for nonlinear system identification, which decouples discrepancy functions from physics-based components. Orthogonal Gaussian process regression balances sparse parameter selection (the white box) with discrepancy learning (the black box) to produce interpretable models from incomplete physics.

Non-stationary partial differential equations (PDEs) arise throughout scientific computing, where the dominant frequency content and energy distribution can drift over time. While efficient in PDE solving, many spectral neural operators apply a shared spectral response across rollout stages, leading to mismatch with time-varying spectra in non-stationary systems. To address this issue, we propose Time-Frequency Gated Spectral Neural Operator (TF-SNO), a state-adaptive framework with learnable time-frequency gating inside spectral blocks. TF-SNO extracts compact frequency-domain and physical-space statistics from the current state to generate modulation coefficients, enabling the spectral response to evolve with the dynamics. TF-SNO learns temporal variation implicitly from the evolving state without introducing an explicit time dimension or time embedding, keeping the modeling complexity low. We further embed the adaptive operator blocks to accurately capture the multi-scale features, thereby improving long-horizon stability. Experiments on six non-stationary PDE benchmarks in 1D and 2D demonstrate that TF-SNO significantly reduces prediction errors and improves robustness compared to strong baselines, with particularly clear gains in long rollout, suggesting the effectiveness of state-dependent spectral adaptation in modeling non-stationary physical systems.

AI-assisted research systems generate many failed attempts, but those failures rarely become a durable, shared knowledge asset. We propose a negative knowledge memory layer: a curator agent converts each failed attempt into a bounded, typed record in a shared bank, and a downstream research agent explicitly adopts or rejects those records before proposing its next experiment. We evaluate this layer in two settings: same-task retry on ScienceAgentBench and cross-task scientific research on two nonlinear math-physics PDE problems. The negative knowledge layer outperforms vanilla AutoResearch baselines while using fewer tokens; agents with the negative knowledge bank solve new tasks that all baselines fail to solve in PDE systems research. We also show that the previous negative knowledge bank can transfer and enhance AutoResearch on different PDE problems. These results suggest that structured negative knowledge is a knowledge asset that should be explicitly maintained in broader AI-engaged scientific research beyond a memory-compression or debugging aid, alongside positive findings, as a collective infrastructure for scientific memory. Code is available at https://github.com/hch-wang/Negative_Knowledge.

Mathematicians understand a PDE solution through mathematical structures rather than tables of computed values. Historically, this has been the product of mathematical analysis, carried out by hand for each problem individually. Neither numerical simulation nor neural networks produce those structures directly. We propose Agentic Symbolic Search (ASYS), a prior-guided framework in which an agent translates PDE theory, public problem constraints, and accumulated search experience into testable differentiable symbolic programs. The mathematical forms are refined under evolutionary search, while their continuous parameters are fit by gradient-based optimization. This makes the search an automated form of inductive-bias injection rather than blind symbolic regression. For problems with known analytical forms, ASYS recovers these forms naturally; for other problems, ASYS constructs analytical approximations which can guide mathematicians toward further analysis. In our experiments, across five problems spanning bounded dynamics, finite-time blow-up, and free-boundary focusing, ASYS produces interpretable representations, including a geometric interface formula for Allen-Cahn 2D dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up, in settings where no closed-form description was previously available. ASYS shows the possibility of a new paradigm for characterizing PDE solutions, beyond handcrafted analytical solutions, mesh-based numerical solutions, and neural network approximations.

Physics-Informed Neural Networks (PINNs) solve Partial Differential Equations (PDEs) by embedding physical laws into neural network training. However, their performance suffers from unstable convergence, training plateaus, and strong sensitivity to architectural and optimization hyperparameters due to the highly non-convex and multi-term structure of the physics-informed loss. In this setting, the outer-loop hyperparameter search is a noisy and black-box optimization problem over heterogeneous parameters, where classical local or gradient-based strategies are easily trapped in suboptimal regions. Evolutionary algorithms, with their population-based exploration and ability to handle mixed, non-differentiable search spaces, provide a more robust mechanism for discovering promising configurations. We propose and investigate a two-stage approach based on evolutionary algorithms that combines exploration and exploitation parts of PINNs training to improve solution accuracy and robustness under fixed computational budgets. In the first stage, we perform low-fidelity training runs with truncated epochs to rapidly screen candidate configurations, treating hyperparameter selection as a black-box outer-loop problem. In the second stage, only the most promising candidates are fully trained with standard gradient-based optimizers to refine the solution. Evaluated on three popular problems, namely Advection, Klein-Gordon and Helmholtz equations, our method consistently outperforms standard training and achieves significantly lower mean error within constrained computational resources.

Inverse problems for differential equations arise throughout science and engineering, where one seeks to infer unknown model parameters from noisy or incomplete observations. Traditional numerical methods for these problems are often computationally expensive, particularly in Bayesian settings where evaluating the likelihood becomes costly for complex forward models and high-dimensional parameter spaces. To address this challenge, we introduce DeepGaLA, a neural-network surrogate for differential equation solvers that provides uncertainty-aware predictions, reducing overconfident inference when training data are limited. To evaluate the fidelity of the surrogate-induced posterior approximations in practice, we show that a short run of delayed-acceptance Markov chain Monte Carlo can serve as an effective diagnostic. Across a range of numerical experiments, DeepGaLA delivers forward-model approximations with accuracy comparable to established Gaussian-process surrogates, while better maintaining efficiency as parameter dimension grows. Moreover, it can incorporate differential-equation constraints, including in nonlinear settings. Overall, these results indicate that uncertainty-quantified neural surrogates can enable scalable and reliable Bayesian inference for inverse problems in complex systems.

We develop QCPIKAN, the first quantum-classical physics-informed Kolmogorov-Arnold network designed to solve partial differential equations (PDEs). Built upon Chebyshev-polynomial KAN layers and parameterized quantum circuits, this hybrid framework embeds physical constraints into the training loss to enforce physical consistency. Our theoretical investigations grounded in approximation theory prove that this design accelerates high-frequency error convergence to an exponential rate and effectively mitigates numerical dispersion. We validate the framework across three typical seepage scenarios in porous media, including single-phase flow, component transport and two-phase flow. Compared with existing quantum-classical physics-informed neural networks, QCPIKAN achieves superior performance in global prediction accuracy, local error control, dynamic evolution tracking and displacement front localization. This work provides a robust and efficient alternative for solving complex PDEs.

Deep learning has managed to evade numerous intuitions from classical statistics to achieve unprecedented performance on a number of real-world tasks. In this article, we investigate the key features and surprises of deep learning from a physics-informed perspective, taking care to point out and justify where possible the many choices inherent in constructing a deep learning model. In particular, we review the phenomenon of neural scaling laws and discuss their interplay with the constraints and inductive biases which may be present when applying machine learning to problems in physics.

Physics-informed neural networks (PINNs) have become a powerful framework for solving PDEs by embedding physical laws into differentiable objectives. Despite their advances, training PINNs remains fragile: recent conflict-averse optimization schemes alleviate gradient interference between residual and boundary losses, but we show that their effectiveness deteriorates as model capacity increases. In this paper, we identify a capacity-induced failure mode, where overparameterized networks undergo functional modularity, self-partitioning into task-exclusive modules that suppress cross-objective interaction and hinder convergence toward Pareto-stationary points. To address this issue, we propose a novel framework, Modular-Sparsity Synchronization (ModSync), which integrates structural optimization into conflict-averse training by penalizing task-exclusive connections while preserving interaction-promoting pathways. Extensive experiments across diverse PDE benchmarks demonstrate that ModSync consistently prevents capacity-driven failures, sustains robust cross-objective coupling, and achieves state-of-the-art accuracy. Codes are available at \url{https://github.com/heejokong/ModSync}.

The matrices arising from large scale $N$-body problems can be efficiently represented using hierarchical matrices, whose key idea is that the admissible off-diagonal sub-matrices can be well approximated by low-rank matrices across a hierarchy of matrix partitions. HODLR (Hierarchical Off-Diagonal Low-Rank) matrices are a subclass of hierarchical matrices in which all off-diagonal submatrices at every level of a recursive binary partition are low-rank. In this article, we present a neural network that learns the inverse operation of HODLR matrices based on the fast direct solver for HODLR matrices developed by Ambikasaran and Darve (2013). We further extend the architecture to learn nonlinear solution operators associated with PDEs by replacing some of the linear layers with deep sub-networks. We demonstrate the performance of the proposed architecture by performing a comprehensive set of experiments that include (i) solving a linear problem such as the Fredholm integral equation of the second kind, (ii) solving PDEs such as the nonlinear Schrödinger equation, Burgers' equation, and the steady-state Darcy's flow equation, (iii) generalization study across varying parameter values, (iv) comparing the inference time of the proposed network with the run time of a classical numerical solver, and (v) comparing the proposed network with some of the existing neural operator learning networks.

We introduce SEA-PINN, a novel architecture that incorporates a Squeeze-Excitation-like attention mechanism into physics-informed neural networks to dynamically recalibrate the importance of neurons across layers. A key feature of SEA-PINN is its highly stable initialization. On 17 out of 20 benchmark problems, SEA-PINN exhibit nearly negligible variance and significantly reduced initial loss, establishing a quasi-deterministic and favorable starting point for optimization. Notably, without employing Fourier feature embeddings or periodic activation functions, SEA-PINN attained competitive accuracy (83% vs. 90% improvement relative to FNN-PINN on the high-frequency case 7) as compared with TSA-PINN-a model specifically engineered for high-frequency problems via learnable frequencies in sinusoidal activations. Furthermore, integrating SEA-PINN into TSA-PINN boosted performance by 42.49%. These results underscore SEA-PINN as a lightweight plug-in module that enhances nonlinear representation power, promotes more robust and efficient convergence, and strengthens the overall reliability of physics-informed learning.

Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems. While traditional numerical solvers are robust, they often incur prohibitive computational costs due to mesh dependencies, whereas recent Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative but frequently suffer from slow convergence and optimization instability. To bridge this gap, this article proposes the Physics-Informed Broad Learning System (PIBLS), a novel backpropagation-free framework that reformulates PDE solving as a direct least-squares optimization. We improved an algorithm within this framework to handle nonlinear PDEs efficiently and provide a rigorous mathematical proof establishing the universal approximation property of PIBLS for these equations. Experiments on linear and nonlinear PDEs demonstrate that PIBLS is one to three orders of magnitude faster than conventional PINNs while achieving significantly higher solution accuracy. This framework provides a computationally efficient paradigm for scientific machine learning, offering a practical, high-speed alternative for real-time simulation and design optimization tasks.

Sparse Mixture-of-Experts (SMoE) architectures are now widely deployed in state-of-the-art language and vision models, where conditional routing allows scaling to very large networks. However, this very Top-$k$ expert selection that enables conditional routing also renders the SMoE map inherently discontinuous. In the vicinity of these discontinuity surfaces, even inputs that are arbitrarily close may activate substantially different sets of experts resulting in significantly different outputs. In this work we give a rigorous geometric and stochastic analysis of these discontinuities. We first classify them by order, determined by the number of tied experts at a switching event. Using measure-theoretic slicing arguments, we establish asymptotic volume estimates for the thickened discontinuity surfaces, showing that lower-order discontinuity sets dominate, whereas higher-order ones occupy a vanishingly small relative volume. Next, modeling random perturbations in the input space via a diffusion process, we prove that the path eventually encounter a discontinuity, and moreover that the first hit almost surely occurs on an order-1 discontinuity with explicit finite-time probability bounds. We further derive occupation-time bounds that quantify the duration the random path spend in the neighborhoods of each discontinuity order. These theoretical results imply that inputs are more likely to lie near lower order discontinuities. Motivated by this insight, we propose a simple smoothing mechanism that can be directly applied to existing SMoEs, softly incorporating experts near discontinuities; our analysis guarantees that the added computational overhead remains small while providing localized smoothing near discontinuities, and experiments across language and vision tasks show that smoothing not only enforces continuity of the SMoE map but also enhances empirical performance.