Partial Differential Equations

The Föllmer process is a Brownian motion conditioned to have a pre-specified distribution at time 1. This process can be interpreted as an ``augmented'' time-compressed version of the reverse stochastic differential equation (SDE) corresponding to the denoising diffusion probabilistic model (DDPM). While this fact has been indirectly used to analyze DDPM sampling errors via discretization of the reverse SDE, the connection between direct discretization of the Föllmer process and the DDPM sampler has not yet been fully explored. This paper clarifies this point while surveying relevant results from the literature. We show that discretized Föllmer processes give natural hyper-parameter settings of the DDPM sampler while accommodating a broader class of variance schedules than discretized reverse SDEs. Moreover, this allows us to systematically recover state-of-the-art results on DDPM sampling error bounds, along with slight improvements.

Accurate simulation of fluid flow in porous media is challenging due to complex pore-space geometries and the computational cost of solving the Navier-Stokes equations. This difficulty is particularly important when repeated simulations are required, as standard numerical solvers may converge slowly in intricate porous domains. We present a neural-network-based framework for predicting pore-scale velocity fields directly from sample geometry. The method uses a convolutional encoder-decoder architecture with skip connections to preserve spatial detail while extracting multi-scale features. Physical consistency is encouraged through a custom loss function combining velocity reconstruction with incompressibility, no-flow conditions inside solids, periodicity constraints, and agreement with the global tortuosity index. We analyze the influence of the corresponding loss weights and quantify the contribution of individual loss components to prediction accuracy. Several CNN backbones are evaluated to identify architectures providing accurate and robust predictions. The generalization ability of the trained model is tested on samples outside the training distribution, including changes in obstacle geometry, boundary conditions, porosity, and realistic porous structures. Finally, we demonstrate a practical use of the predicted velocity fields as initial conditions for Lattice-Boltzmann simulations. This warm-start strategy accelerates solver convergence, reducing the number of iterations in over 90% of tested cases.

Neural operators have emerged as a powerful, discretization-invariant framework for solving partial differential equations (PDEs). Although established approaches like the Deep Operator Network (DeepONet) have successfully achieved universal approximation for operators, and architectures such as Fourier Neural Operators (FNOs) have shown algebraic convergence rates, a precise theoretical connection between the continuous theory and its discrete numerical implementation remains a challenge. Specifically, the relationship between the continuous formulation and the discrete numerical stability has yet to be fully explored. In this paper, we address this gap by establishing theoretical guarantees for the discretization error and stability of neural operator approximation schemes. We prove analytical bounds that link solution regularity to input discretization, providing a formal quantification of neural operator accuracy under real-world numerical constraints. We derive these bounds to the specific cases of State Space Model-based Neural Operators (SS-NOs) and FNOs, thus providing a new discretization error theorem for these models. Additionally, through an input-to-state stability (ISS) analysis, we formally assess the impact of discretization on the stability of SS-NOs results obtained in the continuous domain. Our empirical experiments on 1D and 2D benchmarks validate our theoretical bounds and show the robustness of SS-NOs under varying resolutions.

Digital twins (DTs) rely on continuous synchronization between physical systems and their virtual counterparts through online parameter estimation under uncertainty. In many practical settings, however, this task is challenged by low observability, weak excitation, nonlinear dynamics, and noisy or biased measurements. In this work, we develop a new mathematical framework that integrates Weighted Flow Matching (WFM) generative modeling with physics-informed nonlinear filtering to enhance parameter estimation in DTs. WFM relies on dynamic reweighting of training samples, which guides the generative model toward parameter regimes most informative of the evolving system state. This generative component is tightly coupled with a physics-informed filtering architecture based on the Unscented Kalman Filter (UKF), yielding a unified DT framework that combines data-driven probability transport with physically consistent state and parameter estimation. The effectiveness of the new integrated framework is demonstrated within a spacecraft DT architecture, where stable moment of inertia estimation is achieved under uncertain and noisy sensing, with significant performance improvements over established approaches such as Extended Kalman Filtering (EKF) and Ensemble Kalman Filtering (EnKF). These results highlight the potential of weighted generative modeling as a core mechanism for real-time DT synchronization in operational and mission-critical systems.

Large language models succeed by combining large-scale pretraining with meaningful discrete tokens. In molecular machine learning, SMILES is widely used as a token representation, but it is primarily a linearization format for molecular graphs rather than a semantic decomposition of chemistry. We propose VQ-Atom, a semantic tokenization framework that assigns discrete atom-level tokens based on local chemical environments via vector quantization. Unlike SMILES tokens, VQ-Atom tokens encode graph-local chemical context and are aligned with molecular structure. On protein-cold drug--target interaction prediction using the KIBA dataset, VQ-Atom substantially improves global ranking performance, achieving AUROC of 0.79 while substantially outperforming both SMILES-based and continuous molecular representations under an identical downstream architecture. Furthermore, VQ-Atom enables approximately 3 times faster downstream training than continuous atom-level representations by replacing per-atom continuous features with reusable discrete tokens. These results suggest that molecular tokenization is not merely a preprocessing step, but a central design choice. In particular, well-structured tokens can encode substantial chemical semantics, reducing the burden on downstream learning. VQ-Atom can be interpreted as defining a molecular language, where tokens correspond to chemically meaningful atomic environments, suggesting that token design may constitute an additional axis of machine learning research alongside architecture, objectives, and optimization.

In this paper, we develop a physics-informed deep operator learning framework for solving multi-term time-fractional mixed diffusion-wave equations (TFMDWEs). We begin by deriving an $L_2$ approximation, which achieves first-order accuracy for the Caputo fractional derivative of order $β\in (1,2)$. Building upon this foundation, we propose the fPINN-DeepONet framework, a novel approach that integrates operator learning with the $L_2$ approximation to efficiently solve fractional partial differential equations (FPDEs). Our framework is successfully applied to both fixed and variable fractional-order PDEs, demonstrating the framework's versatility and broad applicability. To evaluate the performance of the proposed model, we conduct a series of numerical experiments that involve dynamically varying fractional orders in both space and time, as well as scenarios with noisy data. These results highlight the accuracy, robustness, and efficiency of the fPINN-DeepONet framework.

Accurate emulation of multi-scale physical systems governed by PDEs demands models that remain stable over long autoregressive rollouts while preserving fine-scale structures. Deterministic emulators produce overly-smoothed predictions, while generative approaches better capture details but are costly. Latent-space generative models have emerged as a compromise but with the additional cost of separately pre-trained autoencoders. We propose Wavelet Flow Matching (WFM), a novel generative emulator that overcomes current trade-offs between cost and skill by performing optimal-transport directly in the multi-scale wavelet space. Rather than learning a latent compression, WFM leverages the hierarchical structure of a U-Net to jointly predict transport velocities of a prescribed wavelet representation. On three challenging systems of chaotic fluid dynamics, WFM achieves superior long-horizon stability, accuracy and spectral coherence compared to state-of-the-art models. Our results clearly position the wavelet space as an effective training-free representation for generative emulation of complex physical dynamics.

Flexible continuous-time survival modeling is critical for capturing complex time-varying hazard dynamics in high-dimensional data; however, training such models remains challenging due to the intractable integral required for likelihood estimation. We introduce QSurv, a scalable deep learning framework that enables nonparametric continuous-time modeling without relying on time discretization or restrictive distributional assumptions. We propose a training objective based on Gauss-Legendre numerical quadrature, which approximates the cumulative hazard with high-order accuracy while facilitating efficient end-to-end training via standard backpropagation. Furthermore, to effectively capture non-stationary hazard dynamics in complex architectures, we introduce time-conditioned low-rank adaptation, a mechanism that conditions general neural backbones on time by dynamically modulating weights via low-rank updates. We provide theoretical analysis establishing approximation error bounds for cumulative-hazard evaluation. Comprehensive experiments across synthetic benchmarks, large-scale real-world tabular datasets, and high-dimensional medical imaging tasks demonstrate that QSurv achieves competitive predictive performance with advantages in instantaneous hazard function estimation, enabling more interpretable characterization of time-varying risk patterns.

The source distribution in conditional flow matching is a design parameter that can be calibrated to data, not a default isotropic prior. We exploit this in Multi-Fidelity Flow Matching (MFFM), a cascade refinement framework for parametric PDE solutions: the source is calibrated to the empirical low-to-high-fidelity residual scale with local Gaussian-blur correlation, and the velocity network is conditioned on the low-fidelity solution. Conditioning makes the residual refinement problem substantially easier than unconditional field generation, while residual-calibrated source noise improves the flow-matching training geometry. A multi-resolution cascade applies the same construction independently between adjacent fidelities. After level-wise flow-matching pretraining, we fine-tune the composed cascade end-to-end with a deterministic one-step rollout, which makes one velocity evaluation per cascade level the optimized operating point at inference. The result is a learned analog of multigrid refinement that reaches the finest grid in $L$ deterministic network evaluations per query. We validate MFFM on eight benchmarks: two super-resolution problems and six spatiotemporal forecasting tasks from PDEBench, The Well, and the FNO Navier--Stokes dataset.

Neural network surrogate models have emerged as a promising approach to model solution fields for a wide variety of boundary value problems encountered in physical modeling. Stochastic problems represent an area of particularly high interest because of the potential to significantly reduce the repeated evaluation of expensive forward models via traditional numerical solvers when conducting parametric analysis. However, many studies found in the literature primarily focus on the ability of neural network surrogate models to represent deterministic samples or mean field solutions and largely overlook surrogate model performance at the tails of the distribution. The present study examines in detail the ability of neural network surrogate models to capture the full distribution of solution fields over the entire probability space, while emphasis is placed at the tails of the distribution. Serving as a canonical problem is the heat conduction equation with a highly stochastic source term, inducing extremely large variation in the thermal solution field. Comparisons are made between a classic feed-forward fully connected network and a Deep Operator Network architecture, using both data-driven and physics-informed loss functions. Results show that the worst-case prediction errors are an order of magnitude larger than the mean field error, highlighting the importance of the outlier samples. The large errors associated with extreme samples result from the networks having to extrapolate beyond the bounds of the training data. A method for identifying these samples is presented along with a discussion of potential approaches to account of their errors. Among the models considered, the fully connected neural network trained using a weak form residual loss performs best in handling these extrapolated inputs, achieving the highest prediction accuracy for the numerically produced datasets.

Obtaining stable diffusion-based samplers in high- and infinite-dimensional settings is challenging because errors can accumulate across high-frequency coordinates and make the dynamics unstable under refinement of the finite-dimensional approximation of the underlying function-space problem. Discretization is a typical source of such errors, and preconditioning with a suitable spectral decay is one way to control their accumulation. In this paper, we study this problem for preconditioned annealed Langevin dynamics (ALD) applied to Gaussian mixtures. We first show that Euler-Maruyama (EM) discretization, by treating the stiff linear part of the annealed score with a forward Euler step, imposes a stability constraint coupling the preconditioner with the annealed covariance scale. Together with the conditions ensuring dimension-uniform control of the annealed dynamics, this constraint forces the initial smoothed law to remain uniformly close to the target across dimensions. We then consider an exponential-integrator scheme that integrates the stiff linear part of the annealed score exactly. Under explicit spectral summability conditions coupling the smoothing covariance, the component covariance spectra, and the preconditioner, we prove a dimension-uniform Kullback-Leibler (KL) bound for this scheme. This bound can be made arbitrarily small, uniformly in dimension, by allowing enough time for annealing and then refining the time mesh accordingly. Importantly, these conditions allow regimes in which the KL divergence between the target and the initial smoothed law diverges with dimension, showing that the restrictions imposed by EM are scheme-dependent rather than intrinsic to ALD.

Monte Carlo methods are essential across diverse scientific fields, yet their efficiency is frequently hampered by critical slowing down-a sharp increase in autocorrelation times near phase transitions. Although deep learning approaches, such as neural-network-based samplers, have been proposed to alleviate this issue, they face another serious problem: the difficulty of training the models. This difficulty partially stems from the overly general nature of original machine-learning architectures, which often ignore underlying physical symmetries and force networks to relearn them from scratch. In this paper, we demonstrate that incorporating physical priors into the model significantly enhances performance. Building upon existing strategies that integrate spin-spin interactions, we propose a framework that utilizes a prior probability distribution as a starting point for training. Our results for the Ising model, as well as for the Edwards-Anderson spin glass model, suggest that moving away from `blank slate' models in favor of physics-informed priors reduces the training burden and facilitates the simulation of larger system sizes in discrete spin models.

Physics-informed neural networks (PINNs) are powerful surrogates for differential equations but are notoriously difficult to train due to spectral bias, stiffness, and poor accuracy on high-frequency or multiscale solutions. Adversarial training based on generative adversarial networks (GANs) has recently gained surprisingly strong empirical results in improving training, but the underlying mechanisms remain elusive. To this end, we propose a new analysis framework for adversarially trained PINNs, based on the key observation of how the discriminator in GANs can influence the training dynamics of PINNs. The framework first provides a much needed theoretical grounding to why and when adversarial training is effective in PINNs, then presents a unified analysis of GANs variants in such training, and finally leads to a new, practical, efficient training algorithm for PINNs. Empirical results demonstrate that our method can significantly reduce the pathology of PINNs training, thereby providing better models with superior performances, often several magnitudes more accurate than alternative methods.

Pre-training neural operators on diverse partial differential equation (PDE) datasets has emerged as a promising direction for building general-purpose surrogate models in scientific machine learning. However, the inherent complexity and structural diversity of PDE solution operators make multi-PDE pre-training fundamentally challenging. Existing methods mainly address this by increasing model capacity, while leaving the target solution operators unchanged. Inspired by classical numerical analysis, we instead propose to transform complex and diverse solution operators into simpler, better-aligned forms that are easier to model jointly. Since the optimal transformation varies across PDE types, it must be adaptive and input-dependent, allowing a single neural operator to approximate an entire family of operators. We instantiate this idea as AOT-POT (adaptive operator-transformation for pre-training operator transformer), which expands hidden representations into multiple parallel streams, adaptively aggregates and redistributes them before and after each sub-layer, and mixes streams through Sinkhorn-projected doubly stochastic matrices for stable training. These mechanisms together reshape diverse solution operators into a unified form that can be effectively modeled by a single architecture. Empirically, AOT-POT achieves state-of-the-art performance on 12 PDE benchmarks with only 3% additional parameters, reducing relative L2 error by up to 77.6% (40.9% on average). Fine-tuning AOT-POT further reduces L2 error by up to 92% on in-domain PDEs and 89% on out-of-domain PDEs (unseen types during pre-training), demonstrating that adaptive operator transformation is an effective and complementary direction for advancing PDE foundation models beyond simply scaling model capacity.

Neural surrogate solvers of partial differential equations (PDEs) promise dramatic speedups over numerical methods, especially in scenarios requiring many solves. However, current accuracy-based evaluations do not fully consider two central issues: (1) neural solvers incur substantial up-front costs for data generation, training, and tuning; and (2) classical solvers can also generate low-fidelity solutions at a sufficiently low simulation cost. To explicitly account for these realities and fully incorporate end-to-end costs, we propose an evaluation framework centered on breakeven complexity, a metric that counts the forward solves before a learned solver is cost-effective relative to an error-equivalent traditional solver. To evaluate this measure, we apply scaling laws to determine how much training budget to allocate to data generation and discuss how to achieve smooth error-matching in diverse settings. We evaluate the breakeven complexity of multiple neural PDE solvers on three PDEs on 2D periodic domains from APEBench and a novel benchmark of flows past multiple obstacles generated by the GPU-native PyFR code. Among other findings, our results suggest that neural PDE solvers become more effective as problems get harder in terms of cost, dimension, rollout, physics regime (e.g. higher Reynolds number), etc.

We introduce Tadpole, a novel foundation model for three-dimensional partial differential equations (PDEs) that addresses key challenges in transferability, scalability to high dimensionality, and multi-functionality. Tadpole is pre-trained as an autoencoder on synthetic 3D PDE data generated by an efficient online data-generation framework. This enables large-scale, diverse training without storage or I/O overhead, demonstrated by scaling to an equivalent of hundreds of terabytes of training data. By autoencoding single-channel spatial crops, Tadpole learns rich and transferable representations across heterogeneous physical systems with varying numbers of state variables and spatial resolutions. Although pre-trained solely as an autoencoder, Tadpole can be efficiently applied for multiple downstream tasks beyond reconstruction, including dynamics learning and generative modeling. For dynamics learning, we propose a novel parameter-efficient fine-tuning strategy that integrates low-rank adaptation, latent-space transformations, and reintroduced skip connections, achieving accurate temporal modeling with a minimal number of trainable parameters. Tadpole demonstrates strong fine-tuning performance across various downstream tasks, highlighting its versatility and effectiveness as a foundation model for 3D PDE learning. Source code and pre-trained weights of Tadpole are available at https://github.com/tum-pbs/tadpole

Physics-Informed Neural Networks (PINNs) combine deep learning with physical constraints for solving partial differential equations (PDEs), and are widely applied in fluid mechanics, heat transfer, and solid mechanics. However, PINN training still suffers from high-dimensional non-convex loss landscapes, imbalanced multiobjective constraints, and ineffective information propagation. Existing curriculum learning and causality-guided strategies improve training stability, but mainly focus on temporal or parametric progression, lacking explicit treatment of spatial information propagation and inter-region consistency. Moreover, they are not directly applicable to boundary value problems (BVPs) with strong spatial coupling. To address this issue, we propose a spatially correlated curriculum learning framework for PINNs. To the best of our knowledge, this is the first work to address PINN training difficulties from the perspective of spatial coupling among subregions. First, spatial causal weights guide information from near-boundary regions inward, reducing optimization failures and spurious convergence. Second, a low-frequency information bridge enforces pseudo-label-based consistency across spatially separated regions, suppressing global low-frequency drift. Third, a region-adaptive reweighting strategy adjusts subregion losses to reduce local residuals and recover high-frequency details. Experiments on PDE benchmarks show that, under comparable computational cost, the proposed method alleviates training failures and improves solution accuracy. The code is available at https://github.com/pigofmomo/CurriculumLearningPINN.

Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics-informed neural networks (PINNs) for solving high-dimensional partial differential equations (PDEs). By leveraging probabilistic representations, BSDE approaches can avoid the curse of dimensionality and often admit second-order-free training objectives that do not require explicit Hessian evaluations. It has recently been established that the commonly used Euler-Maruyama (EM) time discretization induces an intrinsic bias in BSDE training losses. While high-order schemes such as Heun can fully eliminate this bias, such schemes re-introduce second-order spatial derivatives and incur substantial computational overhead. In this work, we provide a principled analysis of EM-induced loss bias and propose an unbiased, second-order-free training framework that preserves the computational advantages of BSDE methods. Our code is available at https://github.com/seojaemin22/Un-EM-BSDE.

Recent advances in neural operators have made partial differential equation (PDE) surrogate modeling increasingly scalable and transferable through large-scale pretraining and in-context adaptation. However, after a shared operator is fine-tuned to multiple regimes within a continuous physical family, it remains unclear whether the resulting weight-space updates merely form isolated regime experts or reveal reusable physical structure. Starting from a shared family anchor, we fine-tune low- and high-regime endpoint experts and show that their updates can be separated into a family-shared adaptation and a direction aligned with the underlying physical parameter. This separation reinterprets endpoint experts as finite-difference probes of a local physical direction in weight space, explaining why static averaging can interpolate between regimes but attenuates endpoint-specific physics. Building on this perspective, we propose Calibration-Conditioned Merge (CCM), a post-hoc coordinate readout method for composing neural PDE experts along this physical direction. Given physical metadata, a calibrated coordinate mapping, or a short observed rollout prefix, CCM infers the target composition coordinate and deploys a single merged checkpoint for the remaining rollout. We evaluate CCM on the reaction--diffusion system, viscosity-parameterized two-dimensional Navier--Stokes equations, and radial dam-break dynamics. Across these benchmarks, CCM achieves its strongest gains in extrapolative regimes, reducing out-of-distribution rollout error relative to the family anchor by 54.2%, 42.8%, and 13.8%, respectively. Further experiments across FNO scales, a DPOT-style backbone, and ablations confirm that endpoint fine-tuning is not arbitrary checkpoint drift, but reveals a calibratable physical direction for training-free transfer across PDE regimes.

We present a physics-informed framework for system identification based on randomized stable atomic features. Impulse responses are represented as random superpositions of stable atoms, namely damped complex exponentials associated with poles sampled inside a prescribed disk. Identification is then cast as a convex regularized least-squares problem with optional linear, second-order-cone, and KYP constraints. The approach generalizes random Fourier and random Laplace features to the damped, nonstationary regime relevant to engineering systems while retaining modal interpretability and scalable finite-dimensional computation. The main analytic point is an operator-theoretic Disk-Bochner viewpoint: positive measures over stable poles generate positive-definite kernels with a radius-dependent shift defect, while a converse scalar disk moment representation for an arbitrary kernel is characterized by subnormality of the canonical shift. We prove this statement, establish an RKHS-to-l1 embedding, show that sampled poles induce a valid finite atomic gauge, discuss random-feature convergence, and state sparse-recovery guarantees conditionally on the restricted-eigenvalue properties of the realized disk-Vandermonde or input-output design matrix. We also connect the normalized transfer function problem to Nevanlinna-Pick interpolation and LFT set-membership. The framework directly encodes stability margins, modal localization, DC-gain bounds, monotonicity, passivity, relative degree, settling-time targets, and time/frequency-domain error bounds. Numerical comparisons illustrate how physically meaningful priors can compensate for poor excitation and improve constrained impulse-response recovery in an under-informative data setting.

Partial differential equations (PDEs) are fundamental for modeling complex natural and physical phenomena. In many real-world applications, however, observational data are extremely sparse, which severely limits the applicability of both classical numerical solvers and existing neural approaches. While neural methods have shown promising results under moderately sparse observations, their inference efficiency at high resolutions is limited, and their accuracy degrades substantially in the extremely sparse regime. In this work, we propose the Di-BiLPS, a unified neural framework that effectively handle both forward and inverse PDE problems under extremely sparse observations. Di-BiLPS combines a variational autoencoder to compress high-dimensional inputs into a compact latent space, a latent diffusion module to model uncertainty, and contrastive learning to align representations. Operating entirely in this latent space, the framework achieves efficient inference while retaining flexible input-output mapping. In addition, we introduce a PDE-informed denoising algorithm based on a variance-preserving diffusion process, which further improves inference efficiency. Extensive experiments on multiple PDE benchmarks demonstrate that Di-BiLPS consistently achieves SOTA performance under extremely sparse inputs (as low as 3%), while substantially reducing computational cost. Moreover, Di-BiLPS enables zero-shot super-resolution, as it allows predictions over continuous spatial-temporal domains.

Physics-Informed Neural Networks (PINNs) and their variational counterparts (VPINNs) are neural networks that incorporate physical laws, making them useful for scientific problems. Existing generalization analyses for PINNs and VPINNs remain limited, often requiring restrictive assumptions such as stability conditions or linear ellipticity. In this paper, we derive generalization bounds for neural networks that involve differentiation with respect to input variables, covering PINNs and VPINNs under a unified framework. We apply Taylor expansion to represent nonlinear differential operators as linear operators on a high-dimensional space, enabling the use of Koopman-based analysis and showing that high-rank networks can generalize well even in settings involving differential operators. We also show that the nonlinearity of the differential operator exponentially enlarges the bound, highlighting its significant impact on generalization.

Determinantal point processes (DPPs) have emerged as a kernelized alternative to vanilla independent sampling for generating efficient minibatches, coresets and other parsimonious representations of large-scale datasets. While theoretical foundations and promising empirical performance have been demonstrated, there are two challenges for current proposals for DPP-based coresets or minibatches. The first is the need for families of DPPs with certain key variance reduction properties, usually constructed in a continuous setting, of which there are few known examples. The second is the need for an ad-hoc construction of a discrete DPP defined on a given dataset, that inherits such variance reduction. In this work, we contribute to the programme of establishing DPPs as a subsampling toolbox for ML by advancing on these two fronts. First, we propose new DPPs on the Euclidean space based on wavelets, with provably better accuracy guarantees than the best known rates. Second, we introduce a general method to convert such continuous DPPs, which are more amenable to proving analytical statements, into discrete kernels, which are pertinent for subsampling tasks such as minibatch and coreset constructions. This conversion mechanism simultaneously preserves the desired variance decay and reveals a low-rank decomposition of the discrete kernel, which makes sampling the corresponding DPP computationally inexpensive. En route, we enlarge the class of ML tasks amenable to improvements via DPP-based minibatches and coresets to include objective functions with arbitrarily low regularity, and rate guarantees that explicitly adapt to this regularity.

Neural operators learn to map initial conditions to the terminal solution of partial differential equations (PDEs), providing a surrogate for the full operator mapping. This enables rapid prediction across different input configurations. While recent neural operator architectures have demonstrated strong performance on diverse PDE tasks, their behavior under structured distribution shifts remains insufficiently understood. To investigate this, we study operator learning in a wave propagation setting governed by a one-dimensional variable-coefficient wave equation, using two representative architectures, the Fourier Neural Operator (FNO) and the Deep Operator Network (DeepONet). To examine their generalization under distribution shifts, we consider structured out-of-distribution (OOD) settings that independently vary input frequency and coefficient smoothness. The results show that under smoothness shifts, both models maintain stable performance, with FNO achieving lower error. In contrast, under frequency shifts, FNO exhibits a sharp increase in error under unseen high-frequency inputs, whereas DeepONet shows milder degradation despite higher overall error. Our analysis reveals that these differences arise from how each architecture represents and responds to variations in frequency structure. Together, these findings highlight a fundamental gap between strong in-distribution performance and generalization under distribution shifts in operator learning, underscoring the role of architectural representation bias in developing more reliable neural operators for physics-based PDE simulations beyond the training distribution.

Solutions to many partial differential equations (PDEs) display coexisting smooth global transport and localized sharp features within a single trajectory: shock fronts, thin interfaces, and concentrated high-frequency content sit on top of slowly varying backgrounds. This poses a challenge for neural operators: Fourier-based architectures mix nonlocal interactions efficiently but tend to under-resolve localized non-smooth features, whereas spatially local architectures recover fine detail at the cost of long-range propagation and rollout stability. Existing hybrid operators paper over this tension with a fixed, spatially uniform fusion that forces the same trade-off everywhere. We propose U-HNO, a U-shaped hybrid neural operator whose central design is Sparse-Point Adaptive Routing (SPAR): at every spatial location, a per-pixel hard mask selects whether the global Fourier branch or the local multi-scale Gaussian branch should dominate, and the sparsity ratio is a function of the local contrast of the routing signal, so smooth and shock-aligned regions receive different mixtures of global and local computation. SPAR is embedded in a hierarchical encoder-bottleneck-decoder backbone with skip connections so that the dual branches and the gate operate at every resolution. Training combines pointwise supervision with a finite-difference H^1 gradient term and a band-wise spectral consistency regularizer. Across benchmarks spanning 1D Burgers, Kuramoto-Sivashinsky, KdV, 2D advection, Allen-Cahn, Navier-Stokes, Darcy flow, and 3D transonic compressible Navier-Stokes from PDEBench, U-HNO achieves state-of-the-art rollout accuracy on the majority of tasks in both relative L^2 and H^1 metrics, with the largest gains on problems dominated by sharp localized features. Ablations show that removing any single component substantially degrades rollout error.

Hybrid machine learning combines physical knowledge with data-driven models to enhance interpretability and performance. In this context, Port-Hamiltonian Systems (PHS), which generalize Hamiltonian mechanics to describe open, non-autonomous dynamical systems, have been successfully integrated with neural networks under the name Port-Hamiltonian Neural Networks (PHNNs). While the ability of PHNNs to identify Hamiltonian ordinary differential equation (ODE) systems has already been demonstrated, their application to learning Hamiltonian partial differential equation (PDE) systems remains largely unexplored. This limitation restricts their use in musical acoustics, where instruments are typically modeled as distributed parameter systems governed by PDEs. In this work, we demonstrate how to learn the nonlinear string dynamics from data in a physically-consistent framework through a PHNN extension to PDEs. By constructing structured neural network architectures based on PHS, we can recover both the Hamiltonian governing the string and the dissipation affecting it. This approach outperforms baseline, non-physics-informed methods in terms of both accuracy and interpretability. Numerical experiments using synthetic data demonstrate the ability of the proposed PHNN model to identify and emulate the nonlinear dynamics of the system.

Neural operators have become an effective framework for learning mappings between function spaces, yet most existing architectures realize operators within a single representational domain, such as physical, spectral, or latent space. In this work, we introduce UFO (Domain-Unification-Free Operator), a cross-domain neural operator framework that realizes operators through adaptive, jointly conditioned interactions among representations defined on distinct domains. UFO enables discretization decoupling: the input function can be observed at resolutions or locations different from those used during training, while the solution can be queried at arbitrary output resolutions. Across four complementary benchmarks covering discontinuous inputs, irregular sampling with spectral mismatch, nonlinear dynamics, and stochastic high-frequency fields, UFO delivers accurate, robust, and physically coherent predictions under distribution shifts. These results establish cross-domain, phase-modulated realization as a powerful framework for discretization-decoupled neural operator learning.

Solving partial differential equations (PDEs) with machine learning typically requires training a new neural network for every new equation. This optimization is slow. We introduce MetaColloc. It is an optimization-free and data-free framework that removes this bottleneck completely. We decouple basis discovery from the solving process. We meta-train a dual-branch neural network on diverse Gaussian Random Fields. This offline process creates a universal dictionary of neural basis functions. At test time, we freeze the network. We solve the PDE by assembling a collocation matrix. We find the solution through a single linear least squares step. For non-linear PDEs, we apply the Newton-Raphson method to achieve fast quadratic convergence. Our experiments across six 2D and 3D PDEs show massive improvements. MetaColloc reaches state-of-the-art accuracy on smooth and non-linear problems. It also reduces test-time computation by several orders of magnitude. Finally, we provide a detailed frequency sweep analysis. This analysis reveals a critical mismatch between function approximation and operator stability at extremely high frequencies. This profound finding opens a clear path toward future operator-aware meta-learning.

Most learned PDE solvers follow a global-surrogate paradigm: a neural operator is trained to map full problem descriptions to full solution fields for a prescribed distribution of geometries, boundary conditions, and coefficients. This has enabled fast inference within fixed problem families, but limits reuse across new domains and makes large-scale deployment dependent on expensive problem-specific data generation. We introduce $\textbf{NEST}$ ($\textbf{Ne}$ural-$\textbf{S}$chwarz $\textbf{T}$iling), a local-to-global framework that shifts learning from full-domain solution operators to reusable local physical solvers. The central premise is that, although global PDE solutions depend on geometry, scale, and boundary conditions, the physical response on small neighborhoods can be learned locally and composed into global solutions through classical domain decomposition. NEST learns a neural operator on minimal voxel patches ($3 \times 3 \times 3$) with diverse local geometries and boundary/interface data. At inference time, an unseen voxelized domain is tiled into overlapping patches, the learned local solver is applied patchwise, and global consistency is enforced through iterative Schwarz coupling with partition-of-unity assembly. In this way, generalization is shifted from a monolithic neural model to the combination of local physics learning and algorithmic global assembly. We instantiate NEST on nonlinear static equilibrium in compressible neo-Hookean solids and evaluate it on large, geometrically complex 3D domains far outside the scale of the training patches. Our results show that local neural building blocks, coupled through Schwarz iteration, offer a reusable local-training path toward scalable learned PDE solvers that generalize across domain size, shape, and boundary-condition configurations.

Diffusion-based posterior sampling (PS) is a leading framework for imaging inverse problems, combining learned priors with measurement constraints. Yet, its standard formulations rely on instantaneous data-consistent estimates, which induce temporal variability in the reverse dynamics. We reinterpret PS from a dynamical perspective, showing that the standard PS update corresponds to a first-order discretization of the diffusion dynamics plus a residual correction capturing the mismatch between the denoised prediction and the data-consistent estimate. A second-order discretization, however, naturally introduces a temporal correction based on the variation of consecutive estimates. Building on this, we propose LAMP, combining the second-order update with the residual correction characterizing a PS technique. LAMP thus inherits a lagged temporal correction, and it can be implemented as a modular plug-in over the PS backbone. We show that LAMP preserves the structure of a posterior sampler, and we perform a one-step risk analysis to characterize when LAMP improves the reverse transition via a bias-variance trade-off. Experiments across multiple imaging tasks demonstrate consistent improvements over strong baselines such as DiffPIR and DDRM, without increasing the number of denoising evaluations.

Most dimensionality reduction methods treat data as discrete point clouds, ignoring the continuous domain structure inherent to many real-world processes. To bridge this gap, we introduce Neural Operator Function Embedding (NOFE), a domain-aware framework for continuous dimensionality reduction. NOFE learns function-to-function mappings via a Graph Kernel Operator, enabling mesh-free evaluation at arbitrary query locations independent of input discretization. We establish NOFE as approximation of sheaf-to-sheaf mappings, generalizing Sheaf Neural Networks to continuous domains. We evaluate NOFE across different datasets, comparing it against PCA, t-SNE, and UMAP. Our results demonstrate that NOFE significantly outperforms baselines in local structure preservation, achieving a local Stress of 0.111 compared to 0.398 for PCA, 0.773 for t-SNE, and 0.791 for UMAP for the ERA5 climate reanalysis dataset. NOFE also exhibits robust sampling independence, reducing the Patch Stitching Error by up to $20.0\times$ relative to UMAP (59.0 vs. 267.6 under regional normalization) and ensuring consistency across disjoint domain patches. While maintaining competitive global structure preservation (Stress-1: 0.379 vs. PCA's 0.268), NOFE resolves fine-grained structures and produces smooth, consistent embeddings that generalize across varying sample densities, addressing key limitations of discrete reduction methods.

Partial differential equations (PDEs) are central to modeling physical and engineering systems, but repeatedly solving parametric PDEs remains computationally expensive. Operator learning enables fast surrogate inference, yet typically requires large input-output paired datasets generated by costly high-fidelity PDE solvers. Unsupervised operator learning frameworks alleviate data dependency but remain hindered by computational bottlenecks. To address this, we propose Neural Variational Quantum Linear Solver (NVQLS), the first hybrid quantum-classical operator learning framework leveraging the Legendre--Galerkin weak formulation. We critically resolve the sign ambiguity in VQLS energy minimization, preventing erroneous solution representations. Additionally, we introduce a neural embedding, a novel encoding scheme to map varying forcings and PDE coefficients into parameterized quantum circuit representations. These structural innovations provide theoretical computational complexity advantages under efficient state preparation schemes, while achieving superior accuracy compared to a representative classical baseline. Validations on 1D and 2D parametric PDEs under diverse boundary conditions demonstrate NVQLS's capability to simultaneously process varying inputs, offering a scalable unsupervised approach to quantum-enhanced operator learning.

Partial differential equations (PDEs) govern diverse physical phenomena, yet high-fidelity numerical solutions are computationally expensive and Machine Learning approaches lack generalization. While Scientific Foundation Models (SFMs) aim to provide universal surrogates, typical encoding-decoding approaches suffer from high pretraining costs and limited interpretability. In this paper, we propose Compositional Neural Operators (CompNO) for 2D systems, a framework that decomposes complex PDEs into a library of Foundation Blocks. Each block is a specialized Neural Operator pretrained on elementary physics. This modular library contains convection, diffusion, and nonlinear convection blocks as well as a Poisson Solver, enabling the framework to address the pressure-velocity coupling. These experts are assembled via an Adaptation Block featuring an Aggregator. This aggregator learns nonlinear interactions by minimizing data loss and physics-based residuals driven from governing equations. The proposed approach has been evaluated on the Convection-Diffusion equation, the Burgers' equation, and the Incompressible Navier-Stokes equation. Our results demonstrate that learning from elementary operators significantly improves adaptability, enhances model interpretability and facilitates the reuse of pretrained blocks when adapting to new physical systems.

Flow matching models generate samples by numerically integrating a learned velocity field, with each integration step requiring a neural network evaluation. Fast generation therefore requires using a small fixed evaluation budget effectively: the key question is not only how to integrate the flow, but where the sampler should spend its steps. We propose SharpEuler, a training-free sampler that profiles a pretrained model offline by estimating where the learned velocity field changes most rapidly along calibration trajectories. This finite-difference estimate defines a solver-aware sharpness profile, which is smoothed and converted by a quantile transform into a timestep grid for any desired inference budget. At test time, sampling remains ordinary Euler integration with the same number of model evaluations as a uniform schedule. We justify SharpEuler using three principles: a numerical principle identifying trajectory acceleration as the leading source of Euler discretization error, a variational principle deriving sharpness-based power-law timestep densities, and a statistical guarantee showing that the finite-sample calibrated sampler is stable at the terminal distribution level. Our experiments show that SharpEuler improves sample quality at fixed budgets, reducing inter-mode leakage and increasing mode coverage.

Data-driven identification of partial differential equations (PDEs) relies on sparse regression over a candidate library of differential operators, where larger libraries inflate false positives under observation noise and smaller libraries risk missing true terms. We introduce Equivariant Operator Discovery (EqOD), a fully automatic method combining two library reduction mechanisms. When Galilean invariance is detected from trajectory data via a weak-form structural test, EqOD uses the symmetry-reduced library, eliminating terms that our Galilean exclusion result proves to be absent from the governing equation. Otherwise, it applies randomized LASSO stability selection guided by classical false-positive bounds. A residual-based fallback prevents degradation below the full-library baseline. On 8 PDEs at 4 noise levels, EqOD attains $F_1 = 1.000 \pm 0.000$ on Heat at $20\%$ noise, where WF-LASSO obtains $0.475 \pm 0.181$, official PySINDy 2.0 obtains $0.000$, and the WSINDy reimplementation obtains $0.789$. Under the strict criterion that the mean F1 difference exceeds the larger of the two standard deviations, EqOD wins 7 of 32 cells. WF-LASSO wins none, and the remaining 25 cells are ties. Across all 32 cells, EqOD outperforms PySINDy 2.0.0 in 23 of 32 cells, and all 5 PySINDy wins occur on reaction PDEs. External validation on WeakIdent and PINN-SR datasets gives $F_1 = 1.000$ on all 5 clean benchmarks. NLS, 2D, coupled-system, and cylinder-wake extensions are reported. The Galilean library reduction is proved under explicit autonomy and library assumptions. The stability-selection step is motivated by classical false-positive bounds, while formal guarantees for correlated PDE design matrices remain open.

Physics-informed deep learning (PIDL) neural networks have shown their capability as a useful instrument for transportation practitioners in utilizing the underlying relationship between the state variables for traffic state estimation (TSE). Another efficient traffic management approach is implementing varying speed limits (VSLs) on transportation corridors to control traffic and mitigate congestion. However, the existing training architecture of PIDL in the literature cannot accommodate the changing traffic characteristics on a freeway with VSL. To tackle this challenge, we propose a novel framework integrating teacher-student ensemble training with PIDL neural networks for TSE under VSL scenarios. The physics of flow conservation law is encoded locally in the teacher models by PIDL, and the student model uses a multi-layer perceptron classifier (MLP) to identify traffic characteristics and selects the ensemble member of PIDL neural networks for TSE. This integrated framework provides a natural solution for capturing the heterogeneity of VSL and accurately addressing the TSE problem. The case study results validate the proposed ensemble approach, demonstrating its superior performance in TSE compared to other popular baseline methods, as indicated by relative L2 error.

The Gauss-Newton matrix is widely viewed as a positive semidefinite approximation of the Hessian, yet mounting empirical evidence shows that Gauss-Newton descent outperforms Newton's method. We adopt a function space perspective to analyze this phenomenon. We show that the generalized Gauss-Newton (GGN) matrix projects the Newton direction in function space onto the model's tangent space, while a Jacobian-only variant obtained by applying the least squares Gauss-Newton matrix to non-least squares losses projects the function space loss gradient onto this same tangent space. Both projections eliminate distortions from the model's parameterization. Specifically, the evolution of the prediction-target mismatch depends on the model's parameterization through the matrix $JJ^\top$ where $J$ is the Jacobian of the model with respect to its parameters. The projections effectively replace $JJ^\top$ with the identity. We call this effect error whitening. Once the parameterization is removed, the prediction-target mismatch evolves according to dynamics dictated by the structure of the loss and the projection produced by the optimizer. Error whitening is a special property of Gauss-Newton descent that rigorously distinguishes it from Newton's method. We empirically demonstrate that Gauss-Newton optimizers follow the theoretically predicted function space dynamics and outperforms Newton's method, Adam, and Muon across case studies spanning supervised learning, physics-informed deep learning, and approximate dynamic programming.

Fast surrogate models for expensive simulations are now essential across the sciences, yet they typically operate as black boxes. We present \texttt{GWAgent}, a large language model (LLM)-based workflow that constructs interpretable analytic surrogates directly from simulation data. Surrogate modeling is well suited to agentic workflows because candidate models can be quantitatively validated against ground-truth simulations at each iteration. As a demonstration, we build a surrogate for gravitational waveforms from eccentric binary black hole mergers. We show that providing the agent with a physics-informed domain ansatz substantially improves output model accuracy. The resulting analytic surrogate attains a median Advanced LIGO mismatch of $6.9\times10^{-4}$ together with an $\sim 8.4\times$ speedup in waveform evaluation, surpassing both symbolic regression and conventional machine learning baselines. Beyond producing an accurate model, the workflow identifies compact physical structure from the learned representation. As an astrophysical application, we use \texttt{GWAgent} to analyze the eccentricity of GW200129 and infer $e_{20\mathrm{Hz}}=0.099^{+0.063}_{-0.044}$. These results show that validation-constrained agentic workflows can produce accurate, fast, and interpretable surrogates for scientific simulations and inference.

We present a framework for distributed Pose Graph Optimization (PGO) by formulating the problem as a second-order continuous-time dynamical system evolving on Lie groups. By modeling pose variables as massive particles subject to damping, the equilibrium points of the resulting Riemannian dynamics coincide with first-order critical points of the original PGO problem. Using the governing damped Euler--Poincaré equations and a semi-implicit geometric integrator, we design an optimization algorithm that generalizes existing algorithms such as Riemannian gradient descent and Gauss--Newton. In multi-robot settings, we present a fully distributed and parallel method based on block-diagonal mass and damping matrices, where each robot solves an ordinary differential equation for its own poses with minimal communication overhead. Moreover, modeling both state and velocity enables principled neighbor prediction that significantly improves convergence under delayed communication. Theoretically, we present an analysis and establish sufficient condition that ensures energy dissipation under the employed geometric discretization scheme. Experiments on benchmark PGO datasets demonstrate that the proposed solver achieves superior performance compared to state-of-the-art distributed baselines in both synchronous and asynchronous regimes.

Scientific methods are developed for classes of problems, so knowledge transfers across structurally related cases. Language-model agents can execute scientific workflows, but their problem--method relationships remain implicit, so each new problem restarts the search and little of what worked transfers. We introduce GRAFT--ATHENA, which makes this problem-to-method map explicit as an expandable probabilistic structure of admissible problems, methods, and their dependencies. Graph factorization keeps the substrate tractable, and semantic fingerprints measure similarity, so experience guides related problems. As a result, the framework matched or exceeded expert baselines, attaining near-machine-precision losses in physics-informed learning, reproducing clinically consistent blood-rheology trends, and developing a high-order hypersonic-flow solver for the Apollo Command Module that matched experimental measurements within $1.8\%$. It also proposed a certified regularization for ill-posed in vivo brain-flow reconstruction, developed a spectrally convergent physics-informed architecture, and established machine-checked universal-approximation theorems for two widely used architectures. Scientific structure enables cumulative and verifiable agentic discovery.

Haptic rendering of viscoelastic materials that exhibit creep and stress relaxation is crucial for many applications, such as medical training with realistic biological tissue models. Fractional-order viscoelastic models provide an effective means of describing intrinsically time-dependent dynamics with few parameters, as these models can naturally capture memory effects. In this study, we present analyses of passivity and rendering performance for fractional-order viscoelastic models under finite-memory discretization. We derive closed-form expressions to ensure the passivity of haptic rendering with a fractional-order (FO) standard linear solid (SLS) model based on Grunwald-Letnikov derivative under short-memory discretization. We also provide symbolic expressions for the effective stiffness and damping of such FO-SLS models. The resulting passivity conditions constitute a unified framework that generalizes previously reported results for integer-order Kelvin-Voigt, Maxwell, and SLS models, since these results are special cases of the newly derived condition. Furthermore, we provide experimental validations of the theoretical passivity bounds and human-subject evaluations of perceived realism of FO-SLS models. Overall, this study establishes a unified theoretical framework and experimental evaluations for FO viscoelastic rendering under short-memory discretization.

Model Predictive Control (MPC) is widely used to operate safety-critical infrastructure by predicting future trajectories and optimizing control actions. However, nonlinear dynamics, hard safety constraints, and numerical optimization often render individual control moves opaque to human operators, undermining trust and hindering deployment. This paper presents Hierarchical Causal Abduction (HCA), which combines (i) physics-informed reasoning via domain knowledge graphs, (ii) optimization evidence from Karush--Kuhn--Tucker (KKT) multipliers, and (iii) temporal causal discovery via the PCMCI algorithm to generate faithful, human-interpretable explanations for control actions computed by nonlinear MPC. Across three diverse control applications (greenhouse climate, building HVAC, chemical process engineering) with expert validation, HCA improves explanation accuracy by 53% over LIME (0.478 vs. 0.311) using a single set of cross-domain parameters without per-domain tuning; domain-specific KKT-threshold calibration over 2--3 days further increases accuracy to 0.88. Ablation studies confirm that each evidence source is essential, with 32--37% accuracy degradation when any component is removed, and HCA's ranking-and-validation methodology generalizes beyond MPC to other prediction-based decision systems, including learning-based control and trajectory planning.

Spectral neural operators achieve strong performance for PDE learning, but rely on fixed global bases that limit their ability to represent spatially heterogeneous and multiscale dynamics. We propose Adaptive Basis Learning (ABLE), a framework that learns data-dependent spectral representations instead of relying on predefined bases. ABLE constructs a spatially adaptive Parseval frame via a learned ancillary density, enabling the operator to act in a lifted spectral space while preserving invertibility and maintaining $O(N\log N)$ complexity through FFT-based implementation. This shifts the source of expressivity from spectral coefficients to the representation itself, allowing the model to capture localized structures and non-translation-invariant interactions more efficiently. ABLE integrates seamlessly into existing neural operator architectures as a drop-in replacement for spectral layers. Across a range of benchmarks ABLE improves accuracy over strong baselines, with the largest gains in regimes characterized by sharp gradients and multiscale behavior. Moreover, augmenting existing models (e.g., U-FNO, HPM) with ABLE further enhances their performance, demonstrating its role as a general and complementary spectral refinement. Our results highlight that the data-driven choice of representation, rather than operator complexity alone, is a key bottleneck in neural operator design. By learning the basis itself, ABLE provides a principled and efficient framework for improving spectral methods in PDE learning.

Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work, as the available information is inherently multifidelity, we propose a kernel learning approach based on cokriging, leveraging empirical information from multifidelity simulations. In the first step, we fit a differentiable non-stationary kernel to an empirical kernel obtained from low-fidelity simulations. In the second step, we derive a high-fidelity kernel with estimated hyperparameters, and construct a corresponding high-fidelity mean using the multifidelity framework. These components can then be used within a Gaussian process framework for solving PDEs. Finally, we demonstrate the performance of the proposed physics-informed method on the Burgers' equation.

Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be discretization-invariant, or resolution-robust, and to reflect PDE-specific structure. It is therefore natural to ask how such structure should be encoded in the model architecture, hypothesis class, or learning procedure. In this paper, we study operator learning for solution operators of nonlinear parabolic PDEs based on Duhamel--Picard iteration. We formulate Picard iteration as an abstract state-transition model and present a theoretical framework for Picard-type operator learning. We derive implementation-agnostic generalization error bounds that separate the implementation error from the estimation error associated with the abstract state-transition model induced by Picard iteration. A key consequence is that increasing the Picard depth reduces the Picard truncation error without causing an unbounded growth of the entropy-based estimation error. We also extend the analysis to long-time prediction by rolling out the same learned local model over successive time blocks. Finally, we illustrate the theory for nonlinear heat equations on the torus using a Picard-type Fourier neural operator as a concrete implementation.

Long-horizon forecasting of time-dependent partial differential equations (PDEs) is critical for characterizing the sustained evolution of physical systems. While neural operators have emerged as efficient surrogates, they typically learn implicit finite-time transitions from discrete observations. When deployed autoregressively, such propagators often suffer from rapid error accumulation and dynamic drift. To address this, we propose a neural forecasting framework that reformulates PDE rollout as learning a Structured Spectral Propagator (SSP) in a propagation-oriented latent space. Following an analysis-propagation-synthesis design, our framework: (i) maps physical states into a shared, time-consistent spatial representation; (ii) projects this space into a compact propagation state to isolate recurrent dynamics from fine-grained spatial details, thereby decoupling reconstruction fidelity from rollout regularity; and (iii) evolves retained spectral modes using a frequency-conditioned linear backbone complemented by a nonlinear spectral closure to account for truncated interactions. This explicit structuring endows the propagator with a strong inductive bias for coherent modal evolution. Extensive experiments demonstrate that SSP significantly outperforms state-of-the-art baselines, reducing relative $L_2$ errors by up to 48.9% and exhibiting improved stability in temporal extrapolation beyond the supervised horizon.

Physics-informed neural networks (PINNs) train a single neural approximation by minimizing multiple physics- and data-derived losses, but the gradients of these losses often interfere and can stall optimization. Existing remedies typically treat this pathology either through scalar loss balancing or full-parameter-space gradient surgery, leaving it unclear which intervention is most appropriate. We show that PINN gradient conflict is not a uniform failure mode with one universal remedy. Instead, we identify distinct PINN gradient-conflict regimes, each associated with a different intervention class. Persistent directional conflict may require separate loss-indexed parameter subspaces, magnitude imbalance often favors scalar reweighting, and low or transient conflict may require no extra mitigation. To select between scalar reweighting and a lightweight architectural intervention, we propose a diagnostic-first framework. It profiles a 1000-step unmodified PINN run and, when intervention is warranted, uses one low-rank adapter per loss to create explicit loss-indexed parameter subspaces attached to a shared PINN trunk, providing each loss with a direct gradient pathway. Across more than 60 PDE configurations, including forward, inverse, multi-physics, parameter-varying, and high-dimensional problems up to 50D, persistent directional conflict dominates standard forward $K=3$ benchmarks and a natural $K=4$ thermoelastic system, where adapters combined with reweighting yield significant improvements. In contrast, $K=3$ inverse problems and natural $K=5$ and $K=6$ multi-physics systems are largely magnitude-dominated and often favor reweighting alone, while full-parameter-space gradient surgery can fail on heterogeneous parameter spaces.

Physics-informed neural networks (PINNs) are a promising approach for solving partial differential equations (PDEs). Their training, however, is often difficult because multiple loss terms induced by PDE residuals and boundary or initial conditions must be optimized simultaneously. To address this difficulty, existing approaches often construct update directions by explicitly enforcing particular desirable properties, such as scale robustness and simultaneous descent. While effective in many cases, such property-by-property designs can make it unclear which conditions are essential, what geometric principle determines the selected update direction, and how different methods are structurally related. In this work, we formulate update-direction selection for PINN training as a Chebyshev-center problem in the dual cone. The proposed formulation selects a normalized direction that maximizes the minimum distance to the cone facets. The resulting formulation admits an efficient dual problem in a much lower-dimensional space and yields a convergence guarantee in the nonconvex setting. It also recovers the key desirable properties targeted by existing approaches without imposing them separately; rather, they follow from the single geometric criterion underlying the formulation. This makes the selected direction interpretable through a single geometric rule and provides a unified basis for systematically comparing related direction-selection methods. Experiments on several PINN benchmarks further demonstrate strong empirical performance of the proposed method.

Selective state space models (SSMs), such as Mamba, achieve strong per-token expressivity by making the time discretization step $\TildeΔ$ a learned function of the input. However, in doing so, $\TildeΔ$ ceases to represent a physical sampling interval, limiting its irregular time series modeling capability. Continuous-time SSMs, such as S5, preserve the physical meaning of $\TildeΔ$ and handle irregular timestamps natively ($\TildeΔ\equivΔ)$, but their dynamics remain linear time-invariant (LTI), limiting per-token expressivity. We propose \textbf{TIDES}, a selective SSM variant that reconciles selective and continuous architectures by moving input-dependence off the step size and onto the diagonal state matrix. As a result, $\TildeΔ$ retains its physical meaning, tied to the state discretization, allowing the model to handle irregular timestamps natively without sacrificing the per-token expressivity that makes selective SSMs effective. We show this on a novel \emph{Fading Flash} experimental benchmark, a compact controlled diagnostic for sequence models that jointly tests input-dependence and extrapolation to out-of-distribution $Δ$ values, and isolates the distinct failure modes of current state-of-the-art architectures that TIDES avoids by construction. On large-scale benchmarks, TIDES sets the new state-of-the-art average rank on UEA time-series classification and the Physiome-ODE regression benchmark. Code available at: https://github.com/TaylanSoydan/TIDES.

Training neural networks to satisfy universal constraints over continuous domains poses unique challenges. Common examples include Lyapunov Neural Networks (Lyapunov NNs) and Physics-Informed Neural Networks (PINNs), where analytical solutions are generally either unavailable or overly restrictive. Sample-based methods are therefore commonly used to enforce these constraints, and the choice of samples has a substantial impact on convergence speed, stability, and solution quality. Most existing methods rely on fixed heuristics or handcrafted rules, and are suboptimal in practice. In this paper, we aim to improve upon them by learning, from data and experience, how to dynamically and iteratively adjust the samples in response to the model's evolving learning performance. Trained by reinforcement learning, the learned policy improves empirical constraint satisfaction on test problems while significantly improving efficiency. We validate the approach on both Lyapunov NNs and PINNs, and demonstrate its broader applicability to domains where adaptive input selection is essential for effective training.

PDE-to-solver code generation aims to automatically synthesize executable numerical solvers from partial differential equation (PDE) specifications. This task requires not only understanding the mathematical structure of PDEs, but also selecting appropriate discretization schemes and solver configurations, and correctly implementing the resulting formulations in finite-element method (FEM) libraries. Existing code generation benchmarks mainly evaluate syntactic correctness, or success on predefined test cases. To our knowledge, there is currently no publicly available benchmark specifically for PDE-to-solver code generation, and general-purpose code benchmarks do not fully capture the unique challenges of numerical PDE solution, such as ensuring solver accuracy, efficiency, and compatibility with professional FEM libraries. We introduce PDEAgent-Bench, to the best of our knowledge, the first multi-metric, multi-library benchmark for PDE-to-solver code generation. PDEAgent-Bench contains 645 instances across 6 mathematical categories and 11 PDE families, with common FEM libraries for DOLFINx, Firedrake, and deal.II. Each instance provides an agent-facing problem specification, a reference solution on a prescribed evaluation grid, and case-specific accuracy and runtime targets. PDEAgent-Bench adopts a staged evaluation framework in which generated solvers must sequentially pass executability, numerical accuracy, and computational efficiency checks. Experiments with representative LLMs and code agents show that models can often produce runnable code, but their pass rate drops substantially once accuracy and efficiency requirements are enforced. These results indicate that current agents remain limited in producing numerically reliable and efficient PDE solvers, and that PDEAgent-Bench provides a reproducible testbed grounded in the practical requirements of numerical PDE solving.

Neural operators provide fast surrogate models for time-dependent partial differential equations, but their standard autoregressive use usually assumes that the instantaneous field $u(t,\cdot)$ is a complete state. This assumption fails for delay equations, distributed-memory systems, and other non-Markovian dynamics: two trajectories may agree at time $t$ and nevertheless have different futures because their histories differ. We introduce the History-Space Fourier Neural Operator (HS-FNO), a neural operator for delay and memory-driven PDEs formulated on the lifted state $u_t(θ,x)=u(t+θ,x)$, $θ\in[-τ,0]$. The key computational step is to decompose one history-state update into a learned predictor for the newly exposed future slice and an exact shift-append transport for the portion of the history window already known from the previous state. This avoids learning deterministic history coordinates, reduces the learned output dimension, and enforces the natural discrete history update. We test HS-FNO on five benchmark families covering delayed reaction--diffusion, spatial epidemiology, nonlocal neural-field dynamics, delayed waves, and distributed-memory closures. Across ten random seeds, HS-FNO attains the lowest aggregate one-step, history-space, and rollout errors among the principal baselines. The largest gain occurs in autoregressive prediction, where aggregate rollout error decreases from $0.241$, $0.188$, and $0.185$ for current-state, lag-stack, and unconstrained history-to-history operators, respectively, to $0.094$. The same model uses fewer parameters than unconstrained history prediction. These results indicate that enforcing the discrete shift structure of history-state evolution is an effective inductive bias for non-Markovian PDE surrogate modeling.

Partial differential equation (PDE) solvers underpin scientific computing, but real-world deployment is bounded by compute. Classical Monte Carlo solvers such as Walk-on-Spheres (WoS) are unbiased and geometry-agnostic but are slow. Learned solvers are fast but biased and brittle under distribution shift. We present \textbf{MC$^2$}, a hybrid WoS-Neural Network (WoS-NN) PDE solver that treats a low-budget Monte Carlo solution as a structured estimator of the true field and learns a single-pass neural correction to recover a high-fidelity solution. MC$^2$ matches the accuracy of solutions using over $1000\times$ more Monte Carlo compute, outperforming all evaluated classical, denoising, and neural-operator baselines. To enable reproducible study of finite-compute PDE solving, we additionally release \textbf{PDEZoo}, the largest standardized elliptic PDE benchmark to date: 2M PDEs spanning five elliptic families and unlimited geometric compositions, with analytic ground truth and multi-budget Monte Carlo trajectories. Together \textbf{MC$^2$} and \textbf{PDEZoo} (1) empirically establish that finite-sample Monte Carlo error is structured, learnable, and correctable in a single forward pass, (2) show that we can solve PDEs $\sim$\textbf{1000x} faster than with just WoS, and (3) provide the evaluation infrastructure the field has so far lacked.

Physics-informed neural networks (PINNs) are a simple surrogate-modelling paradigm for partial differential equations, but their standard strong-form residual formulation is ill suited to the shallow water equations (SWE). It cannot enforce local conservation, handle discontinuities, or leverage the boundary-conforming unstructured meshes used in real-world applications. We introduce ``Data-Guided FVM-PINN'', a framework that replaces the strong-form residual with a differentiable, well-balanced Roe Riemann-solver finite-volume (FVM) loss evaluated on unstructured meshes. The major finding is that physics-only FVM-PINN training often fails on realistic 2D problems: the network collapses to a trivial low-momentum state that nearly satisfies the FVM-PINN residual but bears no resemblance to the true flow. A loss-landscape diagnostic shows that the FVM-PINN loss at zero momentum is only about $7\times$ larger than at the trained solution, a shallow basin that an ordinary optimizer falls into; adding even sparse data turns this into a $310\times$ separation, breaking the degeneracy. On a 2D block-in-channel benchmark, just $200$ random velocity measurements drop the velocity-field $L_2$ error by $22\times$ versus physics-only; $50$ measurements still deliver a $7\times$ reduction. A controlled ablation isolates the contribution of the FVM-PINN loss: it reduces velocity-field $L_2$ by \sim$$23\% in the sparse-data regime and is essentially neutral when dense reference data is available. On a real-world Savannah River reach ($1306$ cells, $3600$~s simulation, five Manning zones), the framework constructs an accurate surrogate from SRH-2D anchor data, with time-window decomposition reducing error monotonically via progressive initial-condition handoff.

Neural operators for time-dependent PDEs face a structural tension: spectral architectures (FNO and descendants) inherit exponential rollout-error growth from their one-step Lipschitz constant, while hierarchical U-Net operators trade resolution invariance for multi-scale detail. We introduce SpectraNet, an autoregressive neural operator that composes truncated spectral convolutions inside a U-Net hierarchy with a Residual-Target Spectral Block trained under a Semigroup-Consistency Loss. The residual-target parametrization replaces L^T stability blow-up with linear T*delta drift, and the spectral path's parameter count is Theta(L w^2 M^2), independent of grid N. Under a single unified protocol against 16 published neural-operator baselines on Navier-Stokes nu=1e-5 at 64x64, SpectraNet reaches test relative L2 = 0.0822 at 2.04M parameters -- 2.33x fewer than canonical FNO at ~20% lower error -- and wins five of six rows in a cross-PDE comparison against FNO (NS at nu in {1e-4, 1e-3}, PDEBench Shallow-Water 2D and Diffusion-Reaction, with the Active-Matter row going to FNO inside its seed spread). Trained from scratch at native 128^2 under the same protocol, SpectraNet improves to 0.0724 while FNO regresses to 0.3080. Free rollout stays bounded for T=100 where FNO diverges across all 200 test trajectories. On consumer CPU at B=1, SpectraNet runs sub-200ms while the full-attention Transformer that wins raw L2 pays ~60x latency; we do not claim to beat that Transformer on raw L2, only to dominate the lightweight (<=5M parameter, sub-200ms CPU) Pareto frontier. Source code: https://github.com/Enrikkk/spectranet

Neural operators have emerged as powerful data-driven solvers for PDEs, offering substantial acceleration over classical numerical methods. However, existing transformer-based operators still face critical challenges when modeling PDEs on complex geometries: directly processing over massive mesh points is computationally expensive, while operating in raw discretization coordinates may obscure the intrinsic geometry where physical interactions are more naturally expressed. To address these limitations, we introduce the Charted Axial Transformer Operator (CATO), a geometry-adaptive and derivative-aware neural operator for PDEs on general geometries. Instead of applying attention directly in the physical coordinate system, CATO learns a continuous latent chart that maps mesh coordinates into a learned chart space, where chart-conditioned axial attention efficiently captures long-range dependencies with reduced computational cost. In addition, CATO introduces a derivative-aware physics loss for steady-state PDEs that jointly supervises solution values, mesh-consistent gradients, and an auxiliary flux-like field, improving physical fidelity and reducing oversmoothing. We further provide a theoretical approximation result showing that, under a favorable chart, charted axial attention can represent low-rank axial solution operators with controlled error, and that small chart perturbations induce bounded approximation degradation. CATO achieves the best performance across all evaluated datasets, yielding an average improvement of approximately 26.76% over the strongest competing baselines while reducing the number of parameters by 81.98%. These results highlight the effectiveness of learning geometry-adaptive charts and derivative-aware physical supervision for accurate and efficient PDE operator learning.

Fourier Neural Operators (FNOs) can greatly accelerate PDE simulation, but they are often used without formal guarantees that they preserve basic physical structure. We show that, once the trained weights and grid are fixed, the spectral convolution in an FNO is a linear map. As a result, the full forward pass is piecewise-linear and can be represented exactly in Z3's linear real arithmetic. We study two encodings. The exact encoding compiles the spectral convolution into a dense matrix multiplication, which is sound for both proofs and counterexamples. The lighter frozen encoding replaces the spectral path with a constant, making it faster but approximate. On 10 small FNO surrogates for 1D advection-diffusion-reaction (85 to 117 parameters, grids 8 to 32), the exact encoding gives 2 sound positivity proofs on linear (ReLU-free) models, 5 sound positivity counterexamples, and 10 sound mass-violation counterexamples; the remaining 3 positivity queries on ReLU models time out. For mass non-increase, Z3 finds worse counterexamples than both gradient-based falsification and Monte Carlo on 7 of 10 models. The frozen encoding scales to grid size 64 with sub-second positivity checks, but it no longer provides certificates for the original FNO. Overall, the results make the soundness--scalability tradeoff explicit and point to what is needed for formal verification of production-scale neural operators.

Deep learning paradigms, such as PINNs and neural operators, have significantly advanced the solving of PDEs. However, they often struggle to capture the continuous integral nature of physical systems, relying either on pointwise residuals that ignore the integral perspective or on pre-discretized temporal grids. Drawing inspiration from MeanFlow, a continuous-time integrator recently developed to efficiently solve generative ODEs, we introduce Spatio-Temporal MeanFlow, which functions as a novel PDE solver learning the finite-interval evolution of physical states. By substituting the generative velocity field with the physical PDE operator, we transform multi-step numerical integration into an efficient prediction with a freely controllable integration length. Crucially, we extend the original MeanFlow constraint from the temporal to the spatio-temporal domain, coupling time evolution with spatial consistency. This yields a unified framework naturally accommodating both time-dependent and stationary PDEs. Comprehensive experiments on benchmarks demonstrate that our approach achieves superior accuracy and inference efficiency over representative baselines. Furthermore, the proposed integral constraint enables excellent generalization to out-of-distribution initial conditions and varying spatial resolutions.

Autoregressive neural simulators now match classical solvers on short-horizon prediction of physical systems, yet their accuracy degrades rapidly when rolled out over long horizons. In this work, we identify transient amplification of perturbations around rollout trajectories as a structural mechanism driving rollout error. Using a linearization analysis we show that when the Jacobians along an autoregressive trajectory are non-normal and non-commuting, the model amplifies errors transiently, resulting in model rollout drift even when the overall system is asymptotically stable. Building on the analysis, we propose commutativity regularization: a combination of two penalties designed to reduce the normality defect of individual Jacobians and the commutator norm of Jacobians across steps. The penalties are estimated with Jacobian-vector products and have no inference-time cost. We show a propagator bound that quantifies rollout error under approximate commutativity and normality. We evaluate UNet and FNO variants with commutativity regularization on 1D and 2D spatio-temporal data in synthetic and real settings, showing successful long-horizon rollouts over thousands of steps. Further, we show that the method improves FourCastNet climate forecasts on ERA5 without using any new data. The gain is most pronounced out-of-distribution: trained on trajectories of a few hundred steps, regularized models remain in-distribution for thousands of rollout steps on initial conditions where baselines diverge.

Neural surrogate models for physical simulations are trained on discretized samples of continuous domains, where the induced empirical measure leads to uneven supervision, biasing optimization and causing spatial inconsistencies in physical fidelity. To mitigate this measure-induced bias, we propose M$^3$ (Multi-scale Morton Measure), a scalable framework that balances training measures by partitioning space according to physical variation and allocating supervision across multiple scales. Applied to three industrial-scale datasets with diverse discretizations, M$^3$ consistently improves predictions in the continuous physical domain, achieving up to 4.7$\times$ lower error in large-scale volumetric cases. These gains persist under aggressive subsampling (160M $\rightarrow$ 16M $\rightarrow$ 1.6M points), where M$^3$-trained models outperform those trained on higher-resolution data, reducing physics-weighted relative $L_2$ error by 3--4$\times$ and the corresponding MSE by up to 13$\times$. These results highlight data distribution as a key factor in operator learning and position M$^3$ as a scalable, data-efficient approach for physically consistent modeling. Code is available at https://github.com/PhysDataRefine/M3.