PDE Operator Learning

PDE: Partial Differential Equation

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Overview

PDE operator learning studies how to approximate solution maps that take functions or physical parameters—such as initial conditions, coefficients, and boundary data—and return PDE solution fields. It aims to make repeated simulation faster for tasks including forecasting, design, inverse problems, and control, using approaches such as Fourier neural operators, DeepONets, attention and graph models, and kernel methods. Researchers also use physics-informed losses, multiscale representations, and structure-preserving constraints to improve accuracy, efficiency, and physical consistency. A central challenge is reliable generalization across unseen geometries, resolutions, frequencies, and parameter regimes while maintaining stable long-term predictions; strong in-distribution accuracy alone does not guarantee it.

Latest papers

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231 papers36 papers in the last four weeks, up 44% on the four weeks before. 0.2% of all new papers.

Apr 22, 2026

Droplet-LNO: Physics-Informed Laplace Neural Operators for Accurate Prediction of Droplet Spreading Dynamics on Complex Surfaces

Spreading of liquid droplets on solid substrates constitutes a classic multiphysics problem with widespread applications ranging from inkjet printing, spray cooling, to biomedical microfluidic systems. Yet, accurate computational fluid dynamic (CFD) simulations are prohibitively expensive, taking more than 18 to 24 hours for each transient computation. In this paper, Physics-Informed Laplace Operator Neural Network (PI-LNO) is introduced, representing a novel architecture where the Laplace integral transform function serves as a learned physics-informed functional basis. Extensive comparative benchmark studies were performed against five other state-of-the-art approaches: UNet, UNet with attention modules (UNet-AM), DeepONet, Physics-Informed UNet (PI-UNet), and Laplace Neural Operator (LNO). Through complex Laplace transforms, PI-LNO natively models the exponential transient dynamics of the spreading process. A TensorFlow-based PI-LNO is trained on multi-surface CFD data spanning contact angles θsε[20,160]θ_s ε[20,160], employing a physics-regularized composite loss combining data fidelity (MSE, MAE, RMSE) with Navier-Stokes, Cahn-Hilliard, and causality constraints.

Ganesh Sahadeo Meshram, Partha Pratim Chakrabarti, Suman Chakraborty

Department of Mechanical Engineering, IIT Kharagpur, Kharagpur, India · Department of Computer Science & Engineering, IIT Kharagpur, Kharagpur, India

PDE Operator LearningPhysics-Informed Neural OperatorsNeural OperatorsComputational Fluid Dynamics
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Apr 20, 2026

Neural Shape Operator Surrogates -- Expression Rate Bounds

We prove error bounds for operator surrogates of solution operators for partial differential and boundary integral equations on families of domains which are diffeomorphic to one common reference (or latent) domain DrefD_{ref}. The pullback of the PDE to DrefD_{ref} via affine-parametric shape encoding produces a collection of holomorphic parametric PDEs on DrefD_{ref}. Sufficient conditions for (uniformly with respect to the parameter) well-posedness are given, implying existence, uniqueness and stability of parametric solution families on DrefD_{ref}. We illustrate the abstract hypotheses by reviewing recent holomorphy results for a suite of elliptic and parabolic PDEs. Quantified parametric holomorphy implies existence of finite-parametric, discrete approximations of the parametric solution families with convergence rates in terms of the number NN of parameters. We obtain constructive proofs of existence of Neural and Spectral Operator surrogates for the shape-to-solution maps with error bounds and convergence rate guarantees uniform on the collection of admissible shapes. We admit principal-component shape encoders and frame decoders. Our results support in particular the (empirically reported) ability of neural operators to realize data-to-solution maps for elliptic and parabolic PDEs and BIEs that generalize across parametric families of shapes.

Helmut Harbrecht, Christoph Schwab

PDE Operator LearningNeural Network Approximation TheoryOperator LearningNeural Operators
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Apr 18, 2026

Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches

This thesis develops numerical and theoretical approaches for understanding and analyzing singularity formation in Partial Differential Equations (PDEs). The singularity formation in the Navier-Stokes Equation (NSE) is famously challenging as one of the seven Clay Prize problems. Unlike simpler equations such as the Nonlinear Heat (NLH) or Keller-Segel (KS) equations, where formal asymptotics near blowup are better understood, the intrinsic complexity of NSE makes quantitative analytical treatment difficult, if not impossible, without numerical guidance. Building on numerical insights, we introduce a robust analytical framework to simplify and systematize pen-and-paper proofs for simpler singular PDEs. We present a novel approach based on enforcing vanishing modulation conditions for perturbations around approximate blowup profiles, complemented by singularly weighted energy estimates. We demonstrate the efficacy of our method on PDEs with complicated asymptotics, such as NLH and the Complex Ginzburg-Landau (CGL) equation, and address the open problem of singularity formation in the 3D KS equation with logistic damping. We develop and refine numerical approaches that facilitate deeper insights into singularity formation. We demonstrate that machine learning methods significantly enhance our capability to identify and characterize potential blowup solutions with high precision. We improve on existing Physics-Informed Neural Network (PINN) and Neural Operator (NO) frameworks. Moreover, we present a novel machine learning paradigm, the Kolmogorov-Arnold Network (KAN) architecture, whose interpretability and excellent scaling properties are achieved through learnable nonlinearities.

Yixuan Wang

CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California · [math.NA]

PDE SolvingKolmogorov-Arnold NetworksPDE Operator LearningPhysics-Informed ML
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Apr 17, 2026

Late Fusion Neural Operators for Extrapolation Across Parameter Space in Partial Differential Equations

Developing neural operators that accurately predict the behavior of systems governed by partial differential equations (PDEs) across unseen parameter regimes is crucial for robust generalization in scientific and engineering applications. In practical applications, variations in physical parameters induce distribution shifts between training and prediction regimes, making extrapolation a central challenge. As a result, the way parameters are incorporated into neural operator models plays a key role in their ability to generalize, particularly when state and parameter representations are entangled. In this work, we introduce the Late Fusion Neural Operator, an architecture that disentangles learning state dynamics from parameter effects, improving predictive performance both within and beyond the training distribution. Our approach combines neural operators for learning latent state representations with sparse regression to incorporate parameter information in a structured manner. Across four benchmark PDEs including advection, Burgers, and both 1D and 2D reaction-diffusion equations, the proposed method consistently outperforms Fourier Neural Operator and CAPE-FNO. Late Fusion Neural Operators achieve consistently the best performance in all experiments, with an average RMSE reduction of 72.9% in-domain and 71.8% out-domain compared to the second-best method. These results demonstrate strong generalization across both in-domain and out-domain parameter regimes.

Eva van Tegelen, Taniya Kapoor, George A. K. van Voorn +2

Artificial Intelligence Group, Wageningen University and Research, Netherlands · Mathematical and Statistical Methods, Wageningen University and Research, Netherlands

Surrogate ModelingPDE Surrogate ModelingOOD GeneralizationPDE Operator LearningNeural Operators
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Apr 7, 2026

Operator Learning for Predicting Bulk Wave Parameters of Spectral Wave Models

The impact of wave-induced forcing on the mean water level and nearshore currents is typically modeled through excess momentum fluxes, also known as radiation stresses, and their spatial gradients. Accurate storm surge prediction requires coupled circulation and wave models, but the high computational cost of numerical wave models limits their temporal resolution. In this work, we explore a proof-of-concept application of Deep Operator Networks (DeepONets) as a surrogate for the Simulating WAves Nearshore (SWAN) numerical wave model. Unlike grid-dependent surrogate models, DeepONets learn the underlying continuous operator, and thus, can provide highly efficient prediction while enabling discretization-invariant inference. The proposed surrogate model is evaluated using two distinct 1-D and 2-D steady-state numerical examples with variable boundary wave conditions and wind fields. When applied to a realistic numerical example of steady-state wave simulation in Duck, NC, the DeepONet surrogate improves computational efficiency by four orders of magnitude. Furthermore, the model demonstrates consistently high accuracy in predicting the significant wave height and the x- and y- components of the radiation stress gradient, by achieving relative L_2 errors bounded by 1.91%, 10.98%, and 6.88%, respectively, across all unseen test scenarios.

Shukai Cai, Sourav Dutta, Mark Loveland +4

Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E 24th St, Austin, 78712, TX, USA · Information Technology Laboratory, U.S. Army Engineer Research and Development Center, MS, USA · Department of Mechanical Engineering and Technology Management, Norwegian University of Life Sciences, Ås, Norway +1

PDE Surrogate ModelingPDE Operator LearningDeep Operator Networks
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Apr 6, 2026

FNO∠θ^{\angle θ}: Extended Fourier neural operator for learning state and optimal control of distributed parameter systems

We propose an extended Fourier neural operator (FNO) architecture for learning state and linear quadratic additive optimal control of systems governed by partial differential equations. Using the Ehrenpreis-Palamodov fundamental principle, we show that any state and optimal control of linear PDEs with constant coefficients can be represented as an integral in the complex domain. The integrand of this representation involves the same exponential term as in the inverse Fourier transform, where the latter is used to represent the convolution operator in FNO layer. Motivated by this observation, we modify the FNO layer by extending the frequency variable in the inverse Fourier transform from the real to complex domain to capture the integral representation from the fundamental principle. We illustrate the performance of FNO in learning state and optimal control for the nonlinear Burgers' equation, showing order of magnitude improvements in training errors and more accurate predictions of non-periodic boundary values over FNO.

Zhexian Li, Ketan Savla

Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089, USA. · Sonny Astani Department of Civil and Environmental Engineering, the Daniel Epstein Department of Industrial and Systems Engineering, and the Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089, USA.

Fourier Neural OperatorPDE Operator LearningOptimal ControlLinear Quadratic Regulator
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Apr 4, 2026

Lotka-Sharpe Neural Operators for Control of Population PDEs

Age-structured predator-prey integro-partial differential equations provide models of interacting populations in ecology, epidemiology, and biotechnology. A key challenge in feedback design for these systems is the scalar ζζ, defined implicitly by the Lotka-Sharpe nonlinear integral condition, as a mapping from fertility and mortality rates to ζζ. To solve this challenge with operator learning, we first prove that the Lotka-Sharpe operator is Lipschitz continuous, guaranteeing the existence of arbitrarily accurate neural operator approximations over a compact set of fertility and mortality functions. We then show that the resulting approximate feedback law preserves semi-global practical asymptotic stability under propagation of the operator approximation error through various other nonlinear operators, all the way through to the control input. In the numerical results, not only do we learn ``once-and-for-all'' the canonical Lotka-Sharpe (LS) operator, and thus make it available for future uses in control of other age-structured population interconnections, but we demonstrate the online usage of the neural LS operator under estimation of the fertility and mortality functions.

Miroslav Krstic, Iasson Karafyllis, Luke Bhan +1

University of California, San Diego, USA · National Technical University of Athens, Greece · KTH Royal Institute Technology, Stockholm, Sweden

Feedback ControlPDE Operator LearningNeural OperatorsDynamical Systems
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Apr 4, 2026

Simple yet Effective: Low-Rank Spatial Attention for Neural Operators

Neural operators have emerged as data-driven surrogates for solving partial differential equations (PDEs), and their success hinges on efficiently modeling the long-range, global coupling among spatial points induced by the underlying physics. In many PDE regimes, the induced global interaction kernels are empirically compressible, exhibiting rapid spectral decay that admits low-rank approximations. We leverage this observation to unify representative global mixing modules in neural operators under a shared low-rank template: compressing high-dimensional pointwise features into a compact latent space, processing global interactions within it, and reconstructing the global context back to spatial points. Guided by this view, we introduce Low-Rank Spatial Attention (LRSA) as a clean and direct instantiation of this template. Crucially, unlike prior approaches that often rely on non-standard aggregation or normalization modules, LRSA is built purely from standard Transformer primitives, i.e., attention, normalization, and feed-forward networks, yielding a concise block that is straightforward to implement and directly compatible with hardware-optimized kernels. In our experiments, such a simple construction is sufficient to achieve high accuracy, yielding an average error reduction of over 17% relative to second-best methods, while remaining stable and efficient in mixed-precision training.

Zherui Yang, Haiyang Xin, Tao Du +1

University of Science and Technology of China · Tsinghua University · Shanghai Qi Zhi Institute

Neural OperatorsPDE Operator LearningTransformer Neural OperatorLow-Rank AttentionLow-Rank Approximation
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Apr 3, 2026

Learning Contractive Integral Operators with Fredholm Integral Neural Operators

We generalize the framework of Fredholm Neural Networks, to learn non-expansive integral operators arising in Fredholm Integral Equations (FIEs) of the second kind in arbitrary dimensions. We first present the proposed Fredholm Integral Neural Operators (FREDINOs), for FIEs and prove that they are universal approximators of linear and non-linear integral operators and corresponding solution operators. We furthermore prove that the learned operators are guaranteed to be contractive, thereby strictly satisfying the mathematical property required for the convergence of the fixed point scheme. Finally, we also demonstrate how FREDINOs can be used to learn the solution operator of non-linear elliptic PDEs, via a Boundary Integral Equation (BIE) formulation. We assess the proposed methodology numerically, via several benchmark problems: linear and non-linear FIEs in arbitrary dimensions, as well as a non-linear elliptic PDE in 2D. Built on tailored mathematical/numerical analysis theory, FREDINOs offer high-accuracy approximations and interpretable schemes, making them well suited for scientific machine learning/numerical analysis computations.

Kyriakos C. Georgiou, Constantinos Siettos, Athanasios N. Yannacopoulos

Department of Electrical Engineering and Information Technologies, University of Naples “Federico II”, Naples, Italy · Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, University of Naples “Federico II”, Naples, Italy

Neural OperatorsPDE Operator Learning
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Apr 1, 2026

PI-JEPA: Label-Free Surrogate Pretraining for Coupled Multiphysics Simulation via Operator-Split Latent Prediction

Reservoir simulation workflows face a fundamental data asymmetry: input parameter fields (geostatistical permeability realizations, porosity distributions) are free to generate in arbitrary quantities, yet existing neural operator surrogates require large corpora of expensive labeled simulation trajectories and cannot exploit this unlabeled structure. We introduce PI-JEPA (Physics-Informed Joint Embedding Predictive Architecture), a surrogate pretraining framework that trains without any completed PDE solves, using masked latent prediction on unlabeled parameter fields under per-sub-operator PDE residual regularization. The predictor bank is structurally aligned with the Lie--Trotter operator-splitting decomposition of the governing equations, dedicating a separate physics-constrained latent module to each sub-process (pressure, saturation transport, reaction), enabling fine-tuning with as few as 100 labeled simulation runs. On single-phase Darcy flow, PI-JEPA achieves 1.9×1.9\times lower error than FNO and 2.4×2.4\times lower error than DeepONet at Nℓ=100N_\ell{=}100, with 24% improvement over supervised-only training at Nℓ=500N_\ell{=}500, demonstrating that label-free surrogate pretraining substantially reduces the simulation budget required for multiphysics surrogate deployment.

Brandon Yee, Pairie Koh

Physics Lab, Yee Collins Research Group · Graduate School of Business, Stanford University

Multiphysics SimulationPDE Surrogate ModelingPDE Operator LearningSelf-Supervised Pre-TrainingJoint-Embedding Predictive ArchitecturePhysics-Based Simulation
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Mar 31, 2026

Lie Generator Networks for Nonlinear Partial Differential Equations

Linear dynamical systems are fully characterized by their eigenspectra, accessible directly from the generator of the dynamics. For nonlinear systems governed by partial differential equations, no equivalent theory exists. We introduce Lie Generator Network-Koopman (LGN-KM), a neural operator that lifts nonlinear dynamics into a linear latent space and learns the continuous-time Koopman generator (LkL_k) through a decomposition Lk=S−DkL_k = S - D_k, where SS is skew-symmetric representing conservative inter-modal coupling, and DkD_k is a positive-definite diagonal encoding modal dissipation. This architectural decomposition enforces stability and enables interpretability through direct spectral access to the learned dynamics. On two-dimensional Navier--Stokes turbulence, the generator recovers the known dissipation scaling and a complete multi-branch dispersion relation from trajectory data alone with no physics supervision. Independently trained models at different flow regimes recover matched gauge-invariant spectral structure, exposing a gauge freedom in the Koopman lifting. Because the generator is provably stable, it enables guaranteed long-horizon stability, continuous-time evaluation at arbitrary time, and physics-informed cross-viscosity model transfer.

Shafayeth Jamil, Rehan Kapadia

Ming Hsieh Department of Electrical and Computer Engineering University of Southern California

Koopman Operator LearningPDE Operator LearningNonlinear System IdentificationComputational Fluid Dynamics
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Mar 26, 2026

DSO: Dual-Scale Neural Operators for Stable Long-term Fluid Dynamics Forecasting

Long-term fluid dynamics forecasting is a critically important problem in science and engineering. While neural operators have emerged as a promising paradigm for modeling systems governed by partial differential equations (PDEs), they often struggle with long-term stability and precision. We identify two fundamental failure modes in existing architectures: (1) local detail blurring, where fine-scale structures such as vortex cores and sharp gradients are progressively smoothed, and (2) global trend deviation, where the overall motion trajectory drifts from the ground truth during extended rollouts. We argue that these failures arise because existing neural operators treat local and global information processing uniformly, despite their inherently different evolution characteristics in physical systems. To bridge this gap, we propose the Dual-Scale Neural Operator (DSO), which explicitly decouples information processing into two complementary modules: depthwise separable convolutions for fine-grained local feature extraction and an MLP-Mixer for long-range global aggregation. Through numerical experiments on vortex dynamics, we demonstrate that nearby perturbations primarily affect local vortex structure while distant perturbations influence global motion trends, providing empirical validation for our design choice. Extensive experiments on turbulent flow benchmarks show that DSO achieves state-of-the-art accuracy while maintaining robust long-term stability, reducing prediction error by over 88% compared to existing neural operators.

Huanshuo Dong, Hao Wu, Hong Wang +2

University of Science and Technology of China, Hefei, China · Tsinghua University, Beijing, China · Institute of Automation, Chinese Academy of Sciences, Beijing, China +1

PDE Operator LearningNeural OperatorsComputational Fluid DynamicsLong-Term Time Series Forecasting
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Mar 25, 2026

Linear-Nonlinear Fusion Neural Operator for Partial Differential Equations

Neural operator learning directly constructs the mapping relationship from the equation parameter space to the solution space, enabling efficient direct inference in practical applications without the need for repeated solution of partial differential equations (PDEs) -- an advantage that is difficult to achieve with traditional numerical methods. In this work, we find that explicitly decoupling linear and nonlinear effects within such operator mappings leads to improved learning efficiency. This yields a novel network structure, namely the Linear-Nonlinear Fusion Neural Operator (LNF-NO), which models operator mappings via the multiplicative fusion of a linear component and a nonlinear component, thus achieving a lightweight and interpretable representation. This linear-nonlinear decoupling enables efficient capture of complex solution features at the operator level while maintaining stability and generality. LNF-NO naturally supports multiple functional inputs and is applicable to both regular grids and irregular geometries. Across a diverse suite of PDE operator-learning benchmarks, including nonlinear Poisson-Boltzmann equations and multi-physics coupled systems, LNF-NO is typically substantially faster to train than several representative neural operator baselines, while achieving comparable or improved accuracy across most tested cases. On the tested 3D Poisson-Boltzmann case, LNF-NO achieves strong accuracy while requiring substantially less training time than the three-dimensional Fourier Neural Operator and Transolver baselines.

Heng Wu, Junjie Wang, Benzhuo Lu

SKLMS, ICMSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China · School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

PDE Operator LearningNeural OperatorsPDE Surrogate Modeling
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Mar 25, 2026

Circuit-Inspired High-Order Neural Networks with Unified Neural Dynamics Modeling for PDE Solving and Visual Perception

Deep networks often rely on architectural heuristics to shape representation evolution, limiting their ability to model data governed by intrinsic dynamics. We present the Circuit-inspired High-Order Neural Network (CHONN), a modular framework that treats representation evolution as a latent potential process and increases its effective order through Kirchhoff-inspired cascade composition. A single Kirchhoff Neural Cell implements a stable first-order update, while serially composed cells form higher-order dynamical operators within one block. This construction is interpretable, numerically stable and compatible with common neural backbones. Theoretical analysis shows that cascaded cells induce end-to-end high-order operators, and controlled experiments demonstrate that intra-block high-order construction differs from generic depth stacking, especially on derivative-sensitive measures. Across steady-state operator learning, long-horizon physical forecasting and ImageNet-1K recognition, CHONN improves structural fidelity, rollout stability and visual representation learning. These results identify high-order circuit composition as a general principle for neural dynamics modeling.

Tongfei Chen, Jingying Yang, Linlin Yang +8

School of Artificial Intelligence (Institute of Artificial Intelligence), Beihang University, Beijing, China. · Communication University of China, Beijing, China. · School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. +3

Dynamic Neural NetworksPDE Operator LearningNeural Operators
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Mar 23, 2026

Adversarial Vulnerabilities in Neural Operator Digital Twins: Gradient-Free Attacks on Nuclear Thermal-Hydraulic Surrogates

Operator learning models are rapidly emerging as the predictive core of digital twins for nuclear and energy systems, promising real-time field reconstruction from sparse sensor measurements. Yet their robustness to adversarial perturbations remains uncharacterized, a critical gap for deployment in safety-critical systems. Here we show that neural operators are acutely vulnerable to extremely sparse (fewer than 1% of inputs), physically plausible perturbations that exploit their sensitivity to boundary conditions. Using gradient-free differential evolution across four operator architectures, we demonstrate that minimal modifications trigger catastrophic prediction failures, increasing relative L2L_2 error from ∼\sim1.5% (validated accuracy) to 37-63% while remaining completely undetectable by standard validation metrics. Notably, 100% of successful single-point attacks pass z-score anomaly detection. We introduce the effective perturbation dimension deffd_{\text{eff}}, a Jacobian-based diagnostic that, together with sensitivity magnitude, yields a two-factor vulnerability model explaining why architectures with extreme sensitivity concentration (POD-DeepONet, deff≈1d_{\text{eff}} \approx 1) are not necessarily the most exploitable, since low-rank output projections cap maximum error, while moderate concentration with sufficient amplification (S-DeepONet, deff≈4d_{\text{eff}} \approx 4) produces the highest attack success. Gradient-free search outperforms gradient-based alternatives (PGD) on architectures with gradient pathologies, while random perturbations of equal magnitude achieve near-zero success rates, confirming that the discovered vulnerabilities are structural. Our findings expose a previously overlooked attack surface in operator learning models and establish that these models require robustness guarantees beyond standard validation before deployment.

Samrendra Roy, Kazuma Kobayashi, Souvik Chakraborty +2

Department of Nuclear, Plasma & Radiological Engineering, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA · Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi 110016, India · National Center for Supercomputing Applications, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA

Adversarial RobustnessNeural OperatorsPDE Operator LearningBlack-Box Adversarial AttacksDigital Twins
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Mar 23, 2026

Generalization Limits of In-Context Operator Networks for Higher-Order Partial Differential Equations

We investigate the generalization capabilities of In-Context Operator Networks (ICONs), a new class of operator networks that build on the principles of in-context learning, for higher-order partial differential equations. We extend previous work by expanding the type and scope of differential equations handled by the foundation model. We demonstrate that while processing complex inputs requires some new computational methods, the underlying machine learning techniques are largely consistent with simpler cases. Our implementation shows that although point-wise accuracy degrades for higher-order problems like the heat equation, the model retains qualitative accuracy in capturing solution dynamics and overall behavior. This demonstrates the model's ability to extrapolate fundamental solution characteristics to problems outside its training regime.

Jamie Mahowald, Tan Bui-Thanh

Department of Mathematics, University of Texas at Austin, Austin, Texas · Department of Aerospace Engineering and Engineering Mechanics, Oden Institute for Computational Engineering and Sciences, University of Texas at Austin, Texas

PDE Operator LearningNeural Network GeneralizationOperator LearningOOD Generalization
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Mar 19, 2026

Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control

We develop deep learning-based approximation methods for fully nonlinear second-order PDEs on separable Hilbert spaces, such as HJB equations for infinite-dimensional control, by parameterizing solutions via Hilbert--Galerkin Neural Operators (HGNOs). We prove the first Universal Approximation Theorems (UATs) which are sufficiently powerful to address these problems, based on novel topologies for Hessian terms and corresponding novel continuity assumptions on the fully nonlinear operator. These topologies are non-sequential and non-metrizable, making the problem delicate. In particular, we prove UATs for functions on Hilbert spaces, together with their Fréchet derivatives up to second order, and for unbounded operators applied to the first derivative, ensuring that HGNOs are able to approximate all the PDE terms. For control problems, we further prove UATs for optimal feedback controls in terms of our approximating value function HGNO. We develop numerical training methods, which we call Deep Hilbert--Galerkin and Hilbert Actor-Critic (reinforcement learning) Methods, for these problems by minimizing the Lμ2(H)L^2_μ(H)-norm of the residual of the PDE on the whole Hilbert space, not just a projected PDE to finite dimensions. This is the first paper to propose such an approach. The models considered arise in many applied sciences, such as functional differential equations in physics and Kolmogorov and HJB PDEs related to controlled PDEs, SPDEs, path-dependent systems, partially observed stochastic systems, and mean-field SDEs. We numerically solve examples of Kolmogorov and HJB PDEs related to the optimal control of deterministic and stochastic heat and Burgers' equations, demonstrating the promise of our deep learning-based approach.

Samuel N. Cohen, Filippo de Feo, Jackson Hebner +1

Mathematical Institute, University of Oxford · Institut für Mathematik, Technische Universität Berlin

HJB EquationsPDE SolvingNeural PDE SolversPDE Operator LearningOptimal ControlNeural Network Approximation Theory+1
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Mar 19, 2026

UniFluids: Unified Neural Operator Learning with Conditional Flow-matching

Partial differential equation (PDE) simulation holds extensive significance in scientific research. Currently, the integration of deep neural networks to learn solution operators of PDEs has introduced great potential. In this paper, we present UniFluids, a conditional flow-matching framework that harnesses the scalability of diffusion Transformer to unify learning of solution operators across diverse PDEs with varying dimensionality and physical variables. Unlike the autoregressive PDE foundation models, UniFluids adopts flow-matching to achieve parallel sequence generation, making it the first such approach for unified operator learning. Specifically, the introduction of a unified four-dimensional spatiotemporal representation for the heterogeneous PDE datasets enables joint training and conditional encoding. Furthermore, we find the effective dimension of the PDE dataset is much lower than its patch dimension. We thus employ xx-prediction in the flow-matching operator learning, which is verified to significantly improve prediction accuracy. We conduct a large-scale evaluation of UniFluids on several PDE datasets covering spatial dimensions 1D, 2D and 3D. Experimental results show that UniFluids achieves strong prediction accuracy and demonstrates good scalability and cross-scenario generalization capability. The code will be released later.

Haosen Li, Qi Meng, Jiahao Li +4

Academy of Mathematics and Systems Science, Chinese Academy of Sciences · Microsoft Research Asia · Renmin University of China +1

PDE Operator LearningPDE Surrogate ModelingConditional Flow Matching
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Mar 18, 2026

RHYME-XT: A Neural Operator for Spatiotemporal Control Systems

We propose RHYME-XT, an operator-learning framework for surrogate modeling of spatiotemporal control systems governed by input-affine nonlinear partial integro-differential equations (PIDEs) with localized rhythmic behavior. RHYME-XT uses a Galerkin projection to approximate the infinite-dimensional PIDE on a learned finite-dimensional subspace with spatial basis functions parameterized by a neural network. This yields a projected system of ODEs driven by projected inputs. Instead of integrating this non-autonomous system, we directly learn its flow map using an architecture for learning flow functions, avoiding costly computations while obtaining a continuous-time and discretization-invariant representation. Experiments on a neural field PIDE show that RHYME-XT outperforms a state-of-the-art neural operator and is able to transfer knowledge effectively across models trained on different datasets, through a fine-tuning process.

Marijn Ruiter, Miguel Aguiar, Jake Rap +2

Digital Futures and Division of Decision and Control Systems, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden. · Control Systems Group, EE Dept., Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

PDE Surrogate ModelingPDE Operator LearningNeural Surrogate ModelingOperator LearningFlow Map Learning
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Mar 18, 2026

Translation Invariance of Neural Operators for the FitzHugh-Nagumo Model

Neural operators (NOs) are powerful deep learning frameworks designed to learn solution operators of partial differential equations. This study evaluates the ability of NOs' to capture the stiff spatio-temporal dynamics of the FitzHugh-Nagumo model. A key contribution of this study is the assessment of the translation invariance using a novel training strategy. Models are trained using an applied current with varying spatial locations and intensities at a fixed time, while the test set presents a challenging out-of-distribution scenario where the current is translated in both time and space. This approach significantly reduces dataset generation costs. We benchmark seven NO architectures: Convolutional Neural Operators (CNOs), Deep Operator Networks (DeepONets), DeepONets with CNN encoders, Proper Orthogonal Decomposition DeepONets, Fourier Neural Operators (FNOs), Tucker Tensorized FNOs, and Local Neural Operators. We evaluated these models based on their accuracy, efficiency, and inference speed. These results demonstrate that CNOs generalize well to translated test dynamics, whereas other architectures do not generalize well. On the training set, all architectures achieve comparable accuracy, with FNOs achieving the highest precision. However, this higher accuracy comes at an elevated computational cost. Meanwhile, DeepONets and their variants exhibit superior training and inference efficiency. These findings highlight the capabilities and limitations of NOs in modeling complex ionic dynamics and provide a comprehensive benchmark for scenarios involving translated dynamics.

Luca Pellegrini

Department of Mathematics, University of Pavia, Pavia, Italy · Euler Institute, Faculty of Informatics, Universit`a della Svizzera italiana, Lugano, Switzerland

Fourier Neural OperatorPDE Operator LearningDeep Operator NetworksNeural Operators
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Mar 16, 2026

Gauge-Equivariant Intrinsic Neural Operators for Geometry-Consistent Learning of Elliptic PDE Maps

Learning solution operators of partial differential equations (PDEs) from data has emerged as a promising route to fast surrogate models in multi-query scientific workflows. However, for geometric PDEs whose inputs and outputs transform under changes of local frame (gauge), many existing operator-learning architectures remain representation-dependent, brittle under metric perturbations, and sensitive to discretization changes. We propose Gauge-Equivariant Intrinsic Neural Operators (GINO), a class of neural operators that parameterize elliptic solution maps primarily through intrinsic spectral multipliers acting on geometry-dependent spectra, coupled with gauge-equivariant nonlinearities. This design decouples geometry from learnable functional dependence and enforces consistency under frame transformations. We validate GINO on controlled problems on the flat torus (T2\mathbb{T}^2), where ground-truth resolvent operators and regularized Helmholtz--Hodge decompositions admit closed-form Fourier representations, enabling theory-aligned diagnostics. Across experiments E1--E6, GINO achieves low operator-approximation error, near machine-precision gauge equivariance, robustness to structured metric perturbations, strong cross-resolution generalization with small commutation error under restriction/prolongation, and structure-preserving performance on a regularized exact/coexact decomposition task. Ablations further link the smoothness of the learned spectral multiplier to stability under geometric perturbations. These results suggest that enforcing intrinsic structure and gauge equivariance yields operator surrogates that are more geometry-consistent and discretization-robust for elliptic PDEs on form-valued fields.

Pengcheng Cheng

School of Mathematics, Jilin University, Changchun, 130012, China

PDE Operator LearningNeural OperatorsPDE SolvingEquivariant Neural Networks
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Mar 15, 2026

Windowed Fourier Propagator: A Frequency-Local Neural Operator for Wave Equations in Inhomogeneous Media

Wave equations are fundamental to describing a vast array of physical phenomena, yet their simulation in inhomogeneous media poses a computational challenge due to the highly oscillatory nature of the solutions. To overcome the high costs of traditional solvers, we propose the Windowed Fourier Propagator (WFP), a novel neural operator that efficiently learns the solution operator. The WFP's design is rooted in the physical principle of frequency locality, where wave energy scatters primarily to adjacent frequencies. By learning a set of compact, localized propagators, each mapping an input frequency to a small window of outputs, our method avoids the complexity of dense interaction models and achieves computational efficiency. Another key feature is the explicit preservation of superposition, which enables remarkable generalization from simple training data (e.g., plane waves) to arbitrary, complex wave states. We demonstrate that the WFP provides an explainable, efficient and accurate framework for data-driven wave modeling in complex media.

Yiyang Cai, Zixuan Qiu, Yunlu Shu +4

School of Mathematical Sciences, Fudan University, Shanghai, 200433, China · Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200438, China · Center for Applied Mathematics, Fudan University, Shanghai, 200438, China

PDE Operator LearningNeural OperatorsNeural Surrogate ModelingFourier Neural Operator
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Mar 14, 2026

Manifold-Orthogonal Dual-spectrum Extrapolation for Parameterized Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have achieved notable success in modeling dynamical systems governed by partial differential equations (PDEs). To avoid computationally expensive retraining under new physical conditions, parameterized PINNs (P2^2INNs) commonly adapt pre-trained operators using singular value decomposition (SVD) for out-of-distribution (OOD) regimes. However, SVD-based fine-tuning often suffers from rigid subspace locking and truncation of important high-frequency spectral modes, limiting its ability to capture complex physical transitions. While parameter-efficient fine-tuning (PEFT) methods appear to be promising alternatives, applying conventional adapters such as LoRA to P2^2INNs introduces a severe Pareto trade-off, as additive updates increase parameter overhead and disrupt the structured physical manifolds inherent in operator representations. To address these limitations, we propose Manifold-Orthogonal Dual-spectrum Extrapolation (MODE), a lightweight micro-architecture designed for physics operator adaptation. MODE decomposes physical evolution into complementary mechanisms including principal-spectrum dense mixing that enables cross-modal energy transfer within frozen orthogonal bases, residual-spectrum awakening that activates high-frequency spectral components through a single trainable scalar, and affine Galilean unlocking that explicitly isolates spatial translation dynamics. Experiments on challenging PDE benchmarks including the 1D Convection--Diffusion--Reaction equation and the 2D Helmholtz equation demonstrate that MODE achieves strong out-of-distribution generalization while preserving the minimal parameter complexity of native SVD and outperforming existing PEFT-based baselines.

Zhangyong Liang, Huanhuan Gao

PDE Operator LearningPhysics-Informed MLOOD GeneralizationParameter-Efficient Fine-Tuning
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Mar 4, 2026

Out-of-distribution transfer of PDE foundation models to material dynamics under extreme loading

Most PDE foundation models are pretrained and fine-tuned on fluid-centric benchmarks. Their utility under extreme-loading material dynamics remains unclear. We benchmark out-of-distribution transfer on two discontinuity-dominated regimes in which shocks, evolving interfaces, and fracture produce highly non-smooth fields: shock-driven multi-material interface dynamics (perturbed layered interface or PLI) and dynamic fracture/failure evolution (FRAC). We formulate the downstream task as terminal-state prediction, i.e., learning a long-horizon map that predicts the final state directly from the first snapshot without intermediate supervision. Using a unified training and evaluation protocol, we evaluate two open-source pretrained PDE foundation models, POSEIDON and MORPH, and compare fine-tuning from pretrained weights against training from scratch across training-set sizes to quantify sample efficiency under distribution shift.

Mahindra Rautela, Alexander Most, Siddharth Mansingh +9

AOT-IC Group, Los Alamos National Laboratory, Los Alamos, New Mexico, US, 87545 · CAI Division, Los Alamos National Laboratory, Los Alamos, New Mexico, US, 87545 · EES Divison, Los Alamos National Laboratory, Los Alamos, New Mexico, US, 87545 +2

PDE Foundation ModelsOOD GeneralizationPDE Operator Learning
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Mar 4, 2026

Jacobian-Adaptive Weighting for Stability: Enhancing Long-term Rollout of Neural Partial Differential Equation Solvers via Spatially-Adaptive Regularization

Data-driven surrogate models can significantly accelerate the simulation of continuous dynamical systems, yet the step-wise accumulation of errors during autoregressive time-stepping often leads to spectral blow-up and unphysical divergence. Existing global regularization techniques can enforce contractive dynamics but uniformly damp high-frequency features, causing over-smoothing; meanwhile, long-horizon trajectory optimization methods are severely constrained by memory bottlenecks. This paper proposes Jacobian-Adaptive Weighting for Stability (JAWS), which reformulates operator learning as a Maximum A Posteriori (MAP) estimation problem with spatially heteroscedastic uncertainty, enabling the regularization strength to adapt automatically based on local physical complexity: enforcing contraction in smooth regions to suppress noise while relaxing constraints near singular features such as shocks to preserve gradient information. Experiments demonstrate that JAWS serves as an effective spectral pre-conditioner for trajectory optimization, allowing short-horizon, memory-efficient training to match the accuracy of long-horizon baselines. Validations on the 1D viscous Burgers' equation and 2D flow past a cylinder (Re=400\text{Re}=400 out-of-distribution generalization) confirm the method's advantages in long-term stability, preservation of physical conservation properties, and computational efficiency. This significant reduction in memory usage makes the method particularly well-suited for stable and efficient long-term simulation of large-scale flow fields in practical engineering applications.

Fengxiang Nie, Yasuhiro Suzuki

University of Hiroshima

Neural PDE SolversAlgorithmic StabilityPDE Operator LearningComputational Fluid Dynamics
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Mar 4, 2026

Extending Neural Operators: Robust Handling of Functions Beyond the Training Set

We develop a rigorous framework for extending neural operators to handle out-of-distribution input functions. We leverage kernel approximation techniques and provide theory for characterizing the input-output function spaces in terms of Reproducing Kernel Hilbert Spaces (RKHSs). We provide theorems on the requirements for reliable extensions and their predicted approximation accuracy. We also establish formal relationships between specific kernel choices and their corresponding Sobolev Native Spaces. This connection further allows the extended neural operators to reliably capture not only function values but also their derivatives. Our methods are empirically validated through the solution of elliptic partial differential equations (PDEs) involving operators on manifolds having point-cloud representations and handling geometric contributions. We report results on key factors impacting the accuracy and computational performance of the extension approaches.

Blaine Quackenbush, Paul J. Atzberger

Department of Mathematics, University of California Santa Barbara (UCSB). · Department of Mathematics, Department of Mechanical Engineering, University of California Santa Barbara (UCSB)

Neural OperatorsOOD GeneralizationPDE Operator LearningNeural Network Approximation Theory
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Mar 3, 2026

Shape Derivative-Informed Neural Operators with Application to Risk-Averse Shape Optimization

Shape optimization under uncertainty (OUU) is computationally intensive for classical PDE-based methods due to the high cost of repeated sampling-based risk evaluation across many uncertainty realizations and varying geometries, while standard neural surrogates often fail to provide accurate and efficient sensitivities for optimization. We introduce Shape-DINO, a derivative-informed neural operator framework for learning PDE solution operators on families of varying geometries, with a particular focus on accelerating PDE-constrained shape OUU. Shape-DINOs encode geometric variability through diffeomorphic mappings to a fixed reference domain and employ a derivative-informed operator learning objective that jointly learns the PDE solution and its Fréchet derivatives with respect to design variables and uncertain parameters, enabling accurate state predictions and reliable gradients for large-scale OUU. We establish a priori error bounds linking surrogate accuracy to optimization error and prove universal approximation results for multi-input reduced basis neural operators in suitable C1C^1 norms. We demonstrate efficiency and scalability on three representative shape OUU problems, including boundary design for a Poisson equation and shape design governed by steady-state Navier-Stokes exterior flows in two and three dimensions. Across these examples, Shape-DINOs produce more reliable optimization results than operator surrogates trained without derivative information. In our examples, Shape-DINOs achieve 3-8 orders-of-magnitude speedups in state and gradient evaluations. Counting training data generation, Shape-DINOs reduce necessary PDE solves by 1-2 orders-of-magnitude compared to a strictly PDE-based approach for a single OUU problem. Moreover, Shape-DINO construction costs can be amortized across many objectives and risk measures, enabling large-scale shape OUU for complex systems.

Xindi Gong, Dingcheng Luo, Thomas O'Leary-Roseberry +2

Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas, USA · School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia · Centre for Data Science, Queensland University of Technology, Brisbane, Australia +3

PDE Operator LearningNeural OperatorsUncertainty QuantificationConstrained Optimization
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Mar 2, 2026

One Operator to Rule Them All? On Boundary-Indexed Operator Families in Neural PDE Solvers

Neural PDE solvers are often described as learning solution operators that map problem data to PDE solutions. In this work, we argue that this interpretation is generally incorrect when boundary conditions vary. We show that standard neural operator training implicitly learns a boundary-indexed family of operators, rather than a single boundary-agnostic operator, with the learned mapping fundamentally conditioned on the boundary-condition distribution seen during training. We formalize this perspective by framing operator learning as conditional risk minimization over boundary conditions, which leads to a non-identifiability result outside the support of the training boundary distribution. As a consequence, generalization in forcing terms or resolution does not imply generalization across boundary conditions. We support our theoretical analysis with controlled experiments on the Poisson equation, demonstrating sharp degradation under boundary-condition shifts, cross-distribution failures between distinct boundary ensembles, and convergence to conditional expectations when boundary information is removed. Our results clarify a core limitation of current neural PDE solvers and highlight the need for explicit boundary-aware modeling in the pursuit of foundation models for PDEs.

Lennon J. Shikhman

College of Computing, Georgia Institute of Technology · Department of Mathematics and Systems Engineering, Florida Institute of Technology

PDE Operator LearningNeural OperatorsNeural Network Generalization
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Mar 1, 2026

Operator Learning Using Weak Supervision from Walk-on-Spheres

Training neural PDE solvers is often bottlenecked by expensive data generation or unstable physics-informed neural network (PINN) involving challenging optimization landscapes due to higher-order derivatives. To tackle this issue, we propose an alternative approach using Monte Carlo approaches to estimate the solution to the PDE as a stochastic process for weak supervision during training. Leveraging the Walk-on-Spheres method, we introduce a learning scheme called Walk-on-Spheres Neural Operator (WoS-NO) which uses weak supervision from WoS to train any given neural operator. We propose to amortize the cost of Monte Carlo walks across the distribution of PDE instances using stochastic representations from the WoS algorithm to generate cheap, noisy, estimates of the PDE solution during training. This is formulated into a data-free physics-informed objective where a neural operator is trained to regress against these weak supervisions, allowing the operator to learn a generalized solution map for an entire family of PDEs. This strategy does not require expensive pre-computed datasets, avoids computing higher-order derivatives for loss functions that are memory-intensive and unstable, and demonstrates zero-shot generalization to novel PDE parameters and domains. Experiments show that for the same number of training steps, our method exhibits up to 8.75×\times improvement in L2L_2-error compared to standard physics-informed training schemes, up to 6.31×\times improvement in training speed, and reductions of up to 2.97×\times in GPU memory consumption. We present the code at https://github.com/neuraloperator/WoS-NO

Hrishikesh Viswanath, Hong Chul Nam, Xi Deng +3

Purdue University, USA · ETH Zurich, Switzerland · California Institute of Technology, USA +1

PDE Operator LearningPhysics-Informed MLNeural OperatorsWeakly Supervised Learning
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Feb 28, 2026

Quantum AS-DeepOnet: Quantum Attentive Stacked DeepONet for Solving 2D Evolution Equations

DeepONet enables retraining-free inference across varying initial conditions or source terms at the cost of high computational requirements. This paper proposes a hybrid quantum operator network (Quantum AS-DeepOnet) suitable for solving 2D evolution equations. By combining Parameterized Quantum Circuits and cross-subnet attention methods, we can solve 2D evolution equations using only 60% of the trainable parameters while maintaining accuracy and convergence comparable to the classical DeepONet method.

Hongquan Wang, Hanshu Chen, Ilia Marchevsky +1

School of Mathematics, Hohai University, Nanjing, Jiangsu 211100, China · College of Mechanics and Engineering Science, Hohai University, Nanjing, Jiangsu 211100, China · Department of Applied Mathematics, Bauman Moscow State Technical University, Moscow 105005, Russia

PDE Operator LearningDeep Operator NetworksHybrid Quantum-Classical ML
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