Neural Operators

Ask about this topic

Overview

Neural operators learn maps from input fields, parameters, or geometries to output fields, often approximating the solution map of a partial differential equation. DeepONet uses branch and trunk networks, Fourier Neural Operators learn interactions in spectral space, and graph and transformer operators handle irregular meshes and complex geometries; physics-informed losses and structure-preserving designs add physical constraints. These models can make repeated simulation, forecasting, and inverse design much faster than running a numerical solver for every case. A central challenge is making predictions reliable across new resolutions, geometries, and physical regimes while preserving stability over long rollouts and matching classical solvers when high accuracy is required.

Latest papers

CardsList

313 papers52 papers in the last four weeks, up 24% on the four weeks before. 0.3% of all new papers.

Sep 2, 2026

Equation Recast for Canonical Operator Learning Across Parametric PDEs

Learning solution operators across broad parameter ranges can require substantial coverage of both input functions and physical parameters, particularly for purely data-driven parametric models. In addition, the resulting models may fail silently outside the training distribution. We introduce equation recast, which reformulates parametric operator learning as the learning of a single canonical operator. Parameter-induced operator variations are derived analytically from the governing equation and absorbed into effective sources, enabling zero-shot prediction across new parameter regimes. Across multi-parameter, nonlinear, and singular PDE settings, equation recast supports extrapolation, integrates sparse heterogeneous datasets in a shared canonical representation, and uses loss of convergence as an internal warning signal for failure of the recast iteration. In high-fidelity tokamak simulations for nuclear fusion, the framework unifies electron-temperature data across four device geometries through canonical-domain mapping within one jointly trained operator. Equation recast provides a route toward reusable neural PDE solvers combining equation-guided transfer, data efficiency, and monitorable inference.

Qiyun Cheng, Valentin Duruisseaux, Cesar F. Clauser +7

1Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA, USA · 2Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA · Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA +1

PDE Operator LearningOperator LearningNeural Operators
AskPDF
Sep 2, 2026

Learning the Constitutive Behavior of Materials via Neural Operators and Causal Attention: Case Studies in Plasticity and Damage

Classical constitutive modeling of path-dependent inelastic materials relies on internal state variables whose evolution equations must be postulated based on domain knowledge and calibrated against experimental data. However, in many practical settings, the relevant internal variables are typically not measurable in experiments, and the constitutive response must be inferred entirely from measured strain-stress data without any prior knowledge of the material's internal state. We propose a data-driven constitutive modeling framework based on the concept of a material operator, which treats a deforming material as a functional mapping from its entire strain history to the corresponding stress response. In contrast to traditional autoregressive or recurrent formulations, the model is trained directly on full loading paths as function-to-function mappings, predicting complete stress trajectories in a single parallel forward pass. Temporal path dependence is enforced through a causally masked attention mechanism embedded within the operator, which restricts the model's attention to past material states while preserving computational parallelizability. Spectral convolutions provide discretization-invariant representations in the frequency domain, while causal attention captures highly adaptive, non-local history dependence. Furthermore, sinusoidal activation functions are used to resolve the strong nonlinear transitions inherent in inelastic regimes. The framework is evaluated across multidimensional, rate-independent material models exhibiting complex phenomena, with an emphasis on nonlinear plasticity and ductile damage accumulation. The results demonstrate accurate and robust predictions of irreversible deformation mechanisms while simultaneously achieving resolution invariance and excellent parallel efficiency.

Rishabh Arora, Lisa Scheunemann, Tim Brepols +1

Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe-Str. 1, D-52074 Aachen, Germany · ACCESS e.V., Intzestr. 5, D-52072 Aachen, Germany

Causal AttentionNeural OperatorsMaterials Science
AskPDF
Sep 1, 2026

Learning with Volterra Neural Networks: A System Theoretic Perspective

Higher-order interaction components are important for signal, image, and video modeling, but explicit high-order operators often suffer from rapidly increasing parameter and computational costs. This paper presents kVNN, a learnable kernelized Volterra Neural operator for compact higher-order filtering. The motivation is to use kernelization to improve the efficiency of Volterra-type neural operators while providing a structured interpretation of their higher-order components. The proposed formulation combines the order-wise structure of Volterra filtering with learnable polynomial-kernel atoms, allowing different interaction orders to be represented by separate learnable centers and coefficients. This order-decoupled representation avoids explicit high-order tensor parameterization and can be implemented as a CNN-compatible layer. Experiments on representative vision tasks show that kVNN achieves a favorable accuracy--efficiency trade-off.

Haoyu Yun, Hamid Krim, Yufang Bao

Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC, USA · Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville, NC, USA

Kernel MethodsNeural Operators
AskPDF
Sep 1, 2026

VATO: A Vortex-Force-Aware Transformer Operator for Unsteady Separated Aerofoil Flows

Accurate prediction of unsteady separated flows is challenging because the aerodynamic loads depend on nonlinear separation and vortex-shedding dynamics. Although high-fidelity CFD resolves these mechanisms, its cost limits repeated use in design and control. Standard field-level surrogate training, however, does not distinguish the flow regions that contribute most strongly to the aerodynamic loads. We introduce VATO (Vortex-Force-Aware Transformer Operator), which couples the Vortex Force Map (VFM) method to a geometry-aware neural operator through two complementary mechanisms. VATO-S adds training-only supervision of the local VFM force-contribution field, with no increase in model size or inference cost. VATO-A uses VFM contribution and sensitivity fields to prioritise force-relevant source locations for residual cross attention. The methods are evaluated on unsteady CFD data for double-edged-plate aerofoils over 54 trajectories from nine geometries. Over lead times of 1-20~ms, VATO-S reduces velocity, pressure, and vorticity errors by 10.4%, 1.0%, and 15.6%, respectively, while VATO-A achieves reductions of 15.8%, 7.5%, and 31.2%. VATO-S gives the lowest VFM-derived drag error, whereas VATO-A gives the lowest pressure-derived lift and drag errors. Over lead times extending 50% beyond the training range, VATO-A retains a 26.9% reduction in vorticity error and larger improvements in all four force readouts, despite reduced gains in velocity and pressure. These results show that force-aware operator learning can improve both flow-field prediction and aerodynamic functional accuracy in unsteady separated flows.

Xingxin Yang, Zhan Zhang, Yichen Li +1

aDepartment of Engineering, King’s College London, London, WC2R 2LS, UK

Transformer Neural OperatorPDE Operator LearningNeural OperatorsComputational Fluid Dynamics
AskPDF
Aug 30, 2026

Joint Spatiotemporal Spectral Neural Operators for Learning PDEs on Irregular Domains

Learning solution operators for partial differential equations (PDEs) on irregular and geometry-dependent domains remains a central challenge in scientific machine learning. While spectral methods provide strong inductive biases for modeling global interactions, they are typically limited to regular domains, and existing neural approaches often require domain warping, interpolation, or costly geometric embeddings. We introduce the Graph Spectral Neural Operator (GSNO), a neural operator that combines spatial graph spectral decompositions with temporal Fourier transforms through a unified space--time spectral kernel. This formulation enables globally coherent operator learning on non-Cartesian discretizations without domain warping or autoregressive rollouts. By replacing learned geometric embeddings with a graph Laplacian spectral basis, GSNO provides geometry-aware spectral learning with low parameter complexity. Across steady and unsteady PDE benchmarks on irregular and geometry-dependent domains, GSNO achieves strong accuracy with reduced runtime and parameter counts, while demonstrating robust zero-shot generalization across mesh resolutions and geometry families.

Abdolmehdi Behroozi, Chaopeng Shen

Department of Civil and Environmental Engineering, Penn State University, University Park, PA 16802, USA

PDE Surrogate ModelingPDE Operator LearningSpectral MethodsNeural Operators
AskPDF
Aug 30, 2026

Sensitivity-Constrained Neural Operators for Data-Efficient Forward and Inverse Modeling of Partial Differential Equation Systems

Neural operators provide fast surrogates for partial differential equation (PDE) solvers, but their reliability can degrade for high-dimensional spatial inputs and inverse or repeated inference. State-only training constrains solution values but not the learned input--output response. We study sensitivity-constrained neural operators (SC-NOs), which augment standard training with sampled solver-derived Jacobian supervision. Selected sensitivities from differentiable solvers or discrete adjoints are matched during training, allowing response information to be amortized across minibatches without imposing the full Jacobian at every update. We evaluate SC-NO on advection--diffusion and RANS--Spalart--Allmaras benchmarks, input-dimensionality scaling tests, long-horizon autoregressive rollout, and a shallow-water Tohoku tsunami source-inversion case. Sensitivity supervision improves forward prediction and yields larger gains in gradient-based inverse reconstruction of distributed fields. Scaling experiments show an improved accuracy--cost tradeoff for high-dimensional gridded inputs, while ablations indicate that state values and Jacobian information provide complementary supervision. In the tsunami case, SC-FNO reconstructs gridded seafloor deformation from sparse early gauge observations and forecasts subsequent wave propagation in a near-real-time proof-of-concept workflow. These results support sampled sensitivity supervision as a practical way to improve neural PDE surrogates when forward accuracy, inverse stability, robustness, and computational cost must be considered together.

Abdolmehdi Behroozi, Chaopeng Shen, Daniel Kifer +1

Department of Civil and Environmental Engineering, Penn State University, University Park, PA 16802, USA · School of Electrical Engineering and Computer Science, Penn State University, University Park, PA 16802, USA

PDE Surrogate ModelingJacobian RegularizationPDE Inverse ProblemsNeural Operators
AskPDF
Aug 28, 2026

Conformal Uncertainty Quantification Guarantees for Neural Operators

Neural operators provide fast surrogate models for approximating operators between function spaces, but their predictions often lack uncertainty quantification. We develop a split conformal framework to guarantee that a calibrated pointwise band around the neural operator output contains the true solution on at least a 1−γ1-γ fraction of the evaluation domain, with probability at least 1−α1-α over test and calibration inputs, where α,γ∈(0,1)α,γ\in(0,1). Our method reduces a normalized residual field to its spatial (1−γ)(1-γ)-quantile and computes a scaling factor using a held-out calibration dataset. We prove marginal coverage guarantees for measurable residual fields defined on arbitrary probability spaces, covering both continuum domains and fixed discretizations. Under mild assumptions on the data distribution, we show that the coverage conditional on the calibration set follows a Beta distribution, which we verify with numerical experiments on Darcy flow and Navier--Stokes equations, where our calibration yields bands consistently tighter than existing corrections while retaining the target coverage.

Tom Stent, Nicolas Boullé

Department of Mathematics Imperial College London London, SW7 2AZ, UK

Fair Conformal PredictionUncertainty QuantificationConformal PredictionNeural Operators
AskPDF
Aug 28, 2026

Euclidean Fourier Neural Operators

Fourier neural operators (FNOs) provide an efficient framework for learning mappings between function spaces as they are, by construction, independent of the grid resolution at which they are trained and evaluated. However, FNOs are not independent of the periodic domain they are applied to: their discrete spectral weights are indexed by integer Fourier mode numbers, which correspond to physical wavevectors. When applied to a different domain, the same trained weights act at different wavevectors, and the FNO silently represents a different operator. This makes FNOs unsuitable for tasks where transfer across domains is crucial. We propose Euclidean Fourier neural operators~(EFNOs) as a domain-independent alternative to FNOs. By parameterizing the spectral kernel as a continuous function of the physical wavevector, the EFNO can learn operators that act consistently across periodic domains of varying shape and size. We evaluate the EFNO on a simple heat equation and on a practically relevant materials science task of learning exchange-correlation potentials across different crystal structures, and demonstrate that the EFNO is able to generalize to unseen grid sizes and domains.

Nathanael Bosch, Niklas Frederik Schmitz, Michael F. Herbst

Mathematics for Materials Modelling (MatMat), Institute of Mathematics & Institute of Materials, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland

Fourier Neural OperatorNeural OperatorsPDE Operator Learning
AskPDF
Aug 27, 2026

Enforcing Dirichlet Boundary Conditions in Operator Learning

Operator learning in scientific machine learning is concerned with approximation of maps between infinite-dimensional function spaces; such maps frequently arise as the solution operators of partial differential equations (PDEs). Neural operators have demonstrated broad empirical success at approximating such maps from data. However, most existing neural operator architectures enforce boundary conditions indirectly through training from data even though the boundary condition is often known exactly. Furthermore, existing modifications and approaches that do enforce boundary conditions explicitly suffer from impractical restrictions, including boundary smoothness, uniform grids, and separable, box-like domains. In this work, we propose an architecture which, independently of training, satisfies homogeneous Dirichlet boundary conditions, whilst simultaneously retaining the expressivity of existing kernel-integral neural operator architectures. This is achieved by enforcing the property that the output of each layer is contained in the span of a subset of the homogeneous Dirichlet eigenfunctions of the Laplacian on the output domain. The method requires only that the output domain be bounded with Lipschitz boundary and places no restriction on the choice of discretization, making it applicable to arbitrary mesh data and general geometries. We prove universal approximation for the resulting architecture; furthermore the approach we adopt in the analysis proves universality for a broad class of kernel-integral neural operators thereby uniting existing theory for a variety of operator learning methods. We validate the proposed method on maps defined by the coefficient to solution map in 2D PDEs: Darcy flow on a square domain and the Helmholtz equation on a circular domain. Comparisons are made with alternative methods.

Andrew M. Stuart, Margaret Trautner

Caltech · ETH Zürich Department of Mathematics

PDE Operator LearningNeural OperatorsPDE SolvingNeural Network Approximation Theory
AskPDF
Aug 26, 2026

Continually learning neural-operator surrogate for three-dimensional airborne electromagnetic Bayesian inversion

Three-dimensional probabilistic inversion of time-domain airborne electromagnetic (AEM) data is limited by the cost of the forward solve. Even though one simulation takes only tens of seconds, a Bayesian inversion of a survey of millions of soundings requires of order 101010^{10} forward evaluations. To address this, we develop a continually learning neural-operator surrogate of the three-dimensional AEM forward operator that replaces the solver inside the Bayesian inversion. We start from the point of view that regardless of what geological prior is specified, Maxwell's laws remain invariant. Secondly, we avoid the limitation of learning on a single prior by continual learning on consecutive priors, which means our surrogate becomes richer as it is applied in future case studies, either by the authors, or by the scientific community. We use a validity check built on ensemble disagreement to divert cases with measurements outside the training range to the solver. Driven by the surrogate, the identical Markov chain Monte Carlo sampler reproduces the full-solver posterior, and its credible intervals cover the truth within 2.6 percentage points. Applied to the 2013 Capricorn TEMPEST survey in Western Australia, the surrogate inverts over two million soundings in seconds, a computation infeasible for the solver. Testing the geological prior against the entire survey costs minutes. The framework delivers uncertainty-quantified conductivity imaging at survey scale, which we believe is essential to perform near real-time mineral-systems targeting with geophysics.

Jaehong Chung, Andrew Lockwood, Jef Caers

Mineral-X, Department of Earth and Planetary Sciences, Stanford University, USA · Xcalibur Smart Mapping, Perth, Australia

PDE Surrogate ModelingBayesian Inverse ProblemsNeural Operators
AskPDF
Aug 26, 2026

A Constitutive Markov Physics-Informed Neural Operator (MPNO) for Autoregressive Stability in Transient Dynamics

Neural operators applied to transient-dynamics PDEs with strong discontinuities exhibit autoregressive instability: in concrete-penetration stress-field prediction, the wavelet neural operator (WNO) diverges in autoregressive rollout, while MeshGraphNets collapse to zero predictions. WNO's instability stems from the lack of a structural constraint on the spectral radius of its propagation operator; the Fourier neural operator (FNO) is stable in these measurements but only emergently, not by construction. We propose a constitutive Markov physics-informed neural operator (MPNO) modeling one-step evolution as a Markov (row-stochastic) propagation operator. Physics-coupled edge weights (acoustic-impedance harmonic mean, contact area, and traction amplitude) encode material-interface constitutive information into a nonnegative symmetric adjacency matrix W; after normalizing the graph Laplacian L = D - W by lambda_max, the propagator P = I - alpha*L~ is constructively constrained to spectral radius rho(P) <= 1, suppressing exponential amplification of autoregressive errors. Stability is thus a designable architectural property, not an optimized loss objective. On three PDEs (Burgers and two-dimensional transverse-section concrete penetration), MPNO rolls out stably with bounded error on all test seeds at 100/135/165 m/s; the single-step relative L2 error is 0.7304 +/- 0.0008, better than WNO and comparable to FNO at about one quarter of FNO's parameters. The edge-weight formula transfers across scenarios by replacing material-property variables. With about 20K parameters, MPNO delivers roughly 10^5x inference speedup over LS-DYNA.

Wenpu Du, Peng Zhou, Yunlong Xia +5

School of Environment and Safety Engineering, North University of China, Taiyuan, 030051, China

PDE Operator LearningPhysics-Informed Neural OperatorsNeural OperatorsPDE Surrogate ModelingDynamical Systems
AskPDF
Aug 25, 2026

Low-Latency Activation-Regularized Sparse Neural Operators with Distillation Assistance Towards Real-Time Neuromorphic Virtual Sensing

Virtual sensing enables digital twins and safety-critical systems to reconstruct and forecast spatial-temporal physics in real time. However, conventional computational and data-driven methods often face challenges in generalization, latency, and energy efficiency for edge deployment. Neural operators offer a promising alternative but remain reliant on power-intensive hardware. Spiking neurons and neuromorphic computing can improve efficiency, yet surrogate-gradient training and multi-step spiking introduce convergence and latency challenges. We propose the Sparse-Activation-ReLU (SAR) layer, a single-step alternative that promotes activation sparsity without surrogate-gradient training while remaining compatible with event-based computing. Within a trunk-based NOMAD architecture, SAR achieves over a fivefold improvement in the combined Latency-Error-Energy (LEE) metric compared with Variable Spiking Neuron (VSN) and Leaky Integrate-and-Fire (LIF) implementations. We further analyze spiking entropy and feature usage and introduce synthetic knowledge distillation, reducing the LEE score by more than twofold. Finally, we improve VSN through a ReLU-based spiking loss and graph-neighbor thresholding. On the Heat Exchanger dataset, these approaches reduce L2 error by more than twofold and nearly sevenfold, respectively, while reducing spiking and spatial aggregation. Overall, the work presented is a step towards energy-efficient virtual sensing by providing an alternative framework that can be positioned towards neuromorphic or other edge device integration that can be a gold standard to compare latency, energy, and error performance for future efficient designs that are sparsity or brain-inspired spiking based.

William Howes, Farid Ahmed, Syed Bahauddin Alam

Grainger College of Engineering, Nuclear, Plasma & Radiological Engineering Department, University of Illinois Urbana-Champaign, Urbana, IL, USA · National Center for Supercomputing Applications, Urbana, IL, USA

Activation SparsityNeuromorphic ComputingNeural OperatorsVirtual Sensing
AskPDF
Aug 24, 2026

Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations

In this paper, we introduce the Inertial Manifold Neural Operator (IMNO) for solving dissipative time-dependent partial differential equations (PDEs). The long-time dynamics of such systems often exhibit an effective low-dimensional structure due to dissipation. Unlike standard neural operator architectures such as the Fourier Neural Operator (FNO), IMNO explicitly leverages the low-dimensional structure to achieve better physical interpretability, accuracy, and stability in long-horizon autoregressive training and prediction for nonlinear dissipative PDEs. For shift-equivariant PDEs, we further introduce a shift-equivariant variant (IMNO-SE) of the proposed neural operator, ensuring that a spatial shift in the input induces the same spatial shift in the output. This symmetry-preserving inductive bias substantially improves its performance in shift-equivariant PDEs. Extensive benchmark experiments are presented to evaluate IMNO's performance numerically.

Xiaoyang Xie, Clarence W. Rowley

Program in Applied and Computational Mathematics (PACM), Princeton University, Princeton, 08540, New Jersey, United States. · Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, 08540, New Jersey, United States.

PDE SolvingPDE Operator LearningNeural Operators
AskPDF
Aug 21, 2026

Read, Write, Relax: Why Neural PDE Surrogates Need Both Global and Local Processing

Recent mesh-based simulation advances have, in no small part, relied on neural surrogates of two distinct families: global models that route information through a small set of latent tokens, and local models that perform message passing across mesh edges. Consistent with both classes is the inability to perform beyond low-dimensional problems and small-scale or oversimplified meshes, the simulation regimes where industrial problems reside. Our work shows this explicitly and presents a unified formulation. In global approaches, latent-token attention acts as a spatial low-pass filter, while local message passing lacks the global reach necessary to propagate information across large mesh spaces. Viewed through the error, the two operators are the halves of a multigrid cycle: one corrects errors at the lower end of the spectrum, the other at the higher end, and neither can do the other's job. We introduce Read-Write-Relax (RWR), which interleaves latent attention with message-passing relaxation under a unified formulation. The interleaved processor lowers error across the entire spectrum, making RWR the most accurate model in nearly every comparison across our industrial and public benchmarks. It is also markedly data-efficient in the scarce-data regimes, accurate on the engineering quantities of interest, and scales full-field predictions to challenging, large-scale problems.

Anuj Kumar, Heiko Zimmermann, Josiah Bjorgaard +4

Pasteur Labs, Brooklyn, NY, USA · Cornell University, Ithaca, NY, USA · Institute for Simulation Intelligence, New York, NY, USA

PDE Surrogate ModelingNeural PDE SolversNeural OperatorsMessage Passing Neural Networks
AskPDF
Aug 20, 2026

STCO: Conditional Neural Operators for Time-Dependent PDEs

Neural operators have emerged as efficient surrogates for time-dependent physical systems governed by partial differential equations (PDEs), but their future-state predictions are often conditioned only on observed states and static problem descriptors. For control or optimization, however, body motion, inflow, or forcing are prescribed for the query without being determined solely by the observed state. We introduce the Spatiotemporal Conditional Operator (STCO) for prescribed-condition operator learning (PCOL), a common interface that supplies prescribed target-time condition fields to heterogeneous backbone architectures while retaining their architecture-specific core computation and context pathways. Its condition interface combines Flow-Aware Graph Leaf (FAGL) with Dual-Site Feature-wise Linear Modulation (DSFiLM). Non-learned FAGL uses vorticity from the final observed frame to construct a fixed-cardinality adaptive partition, then co-locates the observed history and target-time condition fields at its regional coordinates. DSFiLM injects separate motion, inflow, and force routes before and after operator computation through current-feature-driven slot- and channel-wise gates. We evaluate twelve matched backbone architectures with different existing physical and temporal inputs. The immersed-boundary computational fluid dynamics (CFD) benchmark spans prescribed motion, inflow disturbances, body-force actuation, and morphology. Across twelve matched backbones, three regimes, and two lead ranges, STCO yields mean paired reductions of 31.1% in relative-L2 field error and 24.7% in normalized pressure-derived load error. It also lowers longer-lead field error for 11 backbones, while interventions on individual condition groups produce measurable prediction changes for every group evaluated.

Xingxin Yang, Zhan Zhang, Juan Li

Department of Engineering, King’s College London, London, United Kingdom

PDE Operator LearningNeural Operators
AskPDF
Aug 20, 2026

Shared Physics Responses Recover Hidden Rankings in Neural Operator Libraries

Selecting the optimal neural-operator prediction during deployment is challenging when high-fidelity reference solutions are unavailable. We demonstrate that under a squared Hilbert-space loss, ranking a finite model library depends strictly on the low-dimensional span of candidate differences, allowing us to score all models simultaneously using a single anchor-based linearized response of the governing equation. This shared physical diagnostic accurately recovered over 99.6% of pairwise preferences and 99.0% of optimal checkpoints across diverse Fourier and convolutional operator libraries for fluid, reaction-diffusion, and wave dynamics. Furthermore, the corrected physical proxy frequently outperformed the best individual candidates, and we establish computable sufficient conditions that rigorously certify exact decisions for strongly monotone discretizations. By exploiting the local dynamical response rather than raw defect magnitude, this framework enables the reliable and highly efficient deployment of scientific surrogates without requiring ground-truth data.

Hanbing Liang, Fujun Liu

Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology, Changchun 130022, P.R. China

Model SelectionNeural OperatorsPDE Surrogate Modeling
AskPDF
Aug 20, 2026

Wrong-Physics Backdoors in Neural PDE Operators

Neural PDE operators are increasingly trained on reusable solver archives, yet validation often relies on clean prediction error and parameter-agnostic plausibility checks. We introduce cross-parameter relinking, a data-poisoning primitive that makes a triggered input select a valid solution from the same PDE family under an incorrect physical parameter. We term this a wrong-physics backdoor: the output remains physically plausible but is wrong for the intended parameter. The attack exploits tensor-to-parameter provenance failures in multi-parameter archives by stamping the surrogate input and relinking its supervision to a cached alternate-parameter solution for the same latent sample. Across 476 attack campaigns, we evaluate Burgers, advection-diffusion, two-dimensional Navier-Stokes, and an elliptic Poisson case. Fourier Neural Operators and DeepONet provide the primary evidence, with Transformer, GRU, and LSTM models as support. FNO reaches a backdoor success rate of 1.0000 on both advection-diffusion and two-dimensional Navier-Stokes while retaining low clean relative L2 error. Clean-label, label-only, and shuffled controls show that high attack success alone is insufficient: successful attacks must move predictions toward the intended alternate-physics target while preserving bounded clean error. These results expose a structural validation gap: smoothness or generic solver-like behavior is insufficient unless the provenance of the intended physical parameter is also verified.

Hanbing Liang, Fujun Liu

Backdoor AttacksData Poisoning AttacksNeural OperatorsDeep Operator Networks
AskPDF
Aug 19, 2026

Multi-stage neural operator learning with application for convolutions

Convolution integrals widely exist in applications, and to enable fast and accurate computations, this paper introduces two general multi-stage neural operator learning frameworks. The first, Deep Collocation Neural Operator (DCNO), is a supervised approach that iteratively refines the operator approximation by learning residuals from input-output data pairs. The second, Deep Galerkin Neural Operator (DGNO), is an unsupervised framework applicable when the target operator can be represented by a PDE, leveraging the weak form of the PDE residual for training. Both methods progressively construct basis operators through multiple training stages to enrich the approximation space, leading to significantly improved accuracy over standard one-shot operator learning. We provide theoretical analysis for their approximation capabilities and implement them for learning convolutions. Extensive numerical experiments demonstrate that both DCNO and DGNO achieve high accuracy, approaching machine precision under single float for convolution problems, and offer substantial efficiency gains for numerous queries or parametric variations compared to traditional solvers. We also extend these frameworks to handle multi-input operator learning scenarios involving variations in both the density and kernel of a convolution.

Zhiping Mao, Zhenye Wen, Yong Zhang +1

School of Mathematical Sciences, Eastern Institute of Technology, Ningbo, 315200 Ningbo, China · School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China · Center for Applied Mathematics and KL-AAGDM, Tianjin University, 300072 Tianjin, China +2

Operator LearningNeural OperatorsPDE Operator LearningNeural Network Approximation Theory
AskPDF
Aug 19, 2026

GCNO: Gramian Chebyshev Neural Operator for Physics-Based Compression of Wireless Channels

Large antenna arrays allow wireless systems to serve more users and achieve higher data rates, but they also make channel feedback expensive: the receiving device must repeatedly report a large complex-valued channel matrix to the base station. Most neural compressors treat this matrix like an image and replace it with a fixed-length code that only a matched neural decoder can interpret. The message therefore does not adapt to channel complexity, and changing the antenna count typically requires retraining. We ask whether a device can instead report only the few dominant propagation paths underlying each channel. We introduce the Gramian Chebyshev Neural Operator (GCNO), a physics-based, variable-rate compressor that identifies a sample-dependent set of path directions. GCNO uses receive-transmit channel structure to locate paths, a first-order Taylor correction to refine directions that fall between grid points, and least squares to recover their complex strengths. It is trained without path labels, and the base station reconstructs the channel analytically from the transmitted path tuples rather than through a learned decoder. Across three ray-traced environments, GCNO achieves better reconstruction accuracy at the same payload - or lower payload at the same accuracy - than neural feedback baselines, and transfers to unseen antenna counts without retraining.

Rafid Umayer Murshed, Shahab Hamidi-Rad, Elahe Soltanaghai +1

Department of Computer Science, University of Illinois Urbana-Champaign · InterDigital AI Lab

Wireless CommunicationsNeural OperatorsSparse Recovery
AskPDF
Aug 18, 2026

Self-supervised In-context Operator Learning for Stochastic Mean-Field Control

Stochastic mean-field control (MFC) provides a fundamental framework for coordinating large populations of interacting agents under uncertainty, with a wide range of applications. Existing numerical and deep-learning methods solve one MFC problem instance at a time and must be re-optimized whenever the task changes. In this work, we formulate stochastic MFC as an operator-learning problem and develop, to the best of our knowledge, the first mesh-free, self-supervised neural operator for stochastic MFC. The main challenge is that the diffusion term in the controlled Fokker--Planck equation precludes deterministic transport-map representations. We address this challenge by combining the probability-flow ODE with an invertible normalizing-flow-based transformer, which recasts the dynamics as a deterministic continuity equation and enables closed-form score evaluation through the exact inverse and analytical log-determinant of the normalizing flow, with O(d)\mathcal{O}(d) cost per particle for networks of fixed size. Through transformer-based in-context learning, task prompts, represented by compact distribution parameters or raw particle clouds, condition the transport map, enabling a single pretrained operator to solve unseen tasks in one forward pass. The resulting Normalizing Flow Invertible Solution Transformer (NFIST) is trained end-to-end by minimizing the stochastic control objective directly, requiring no precomputed numerical solutions for training. We further prove the consistency of the proposed operator-learning formulation with task-by-task optimization. Numerical experiments on stochastic optimal control, Schrödinger bridge, systemic-risk control, and obstacle-avoiding path planning demonstrate effective zero-shot generalization while substantially reducing the computational cost of solving large families of stochastic MFC problems.

Suyi Gao, Mo Zhou, Rongjie Lai

Department of Mathematics, Purdue University, West Lafayette, USA · Department of Mathematics, University of California, Los Angeles, USA

Operator LearningNeural OperatorsStochastic Optimal Control
AskPDF
Aug 15, 2026

MiNO: Cotangent-bundle propagator learning for PDEs

Scientific machine learning for partial differential equations commonly targets solution fields, as in physics-informed neural networks, or solution maps, as in neural operators. We study a third target: the propagator itself, a phase and amplitude in phase space. The motivation is a gap in regularity. A transported discontinuity is nonsmooth in space and time, yet the rule that moves it can be a polynomial phase carrying unit amplitude, so the object that generates an evolution can be far smoother than the field it generates. The microlocal neural operator (MiNO) learns that object, using the eikonal equation for the phase and the transport equation for the amplitude, and recovers the solution by an oscillatory integral. Sharp fronts and caustics then belong to propagation geometry rather than to a field fitted pointwise. Small residuals certify more than the reconstructed field. They place the learned canonical relation, the geometry that carries singularities, close to the exact one, and they separate trainable error from the frequency-truncation tail. On a matched-budget discontinuous-advection benchmark, MiNO stops improving within 10,000 steps at the accuracy limit of its finite reconstruction window, a limit predicted in closed form, whereas a physics-informed neural network with neural-tangent-kernel loss balancing stays near its initial error. On smooth advection, the mean error is 3.84×10−33.84\times10^{-3} for MiNO and 3.12×10−23.12\times10^{-2} for a supervised Fourier neural operator. Single-branch MiNO is the smallest model compared, and one trained generator serves five unseen initial conditions without retraining.

Gnankan Landry Regis N'guessan, Bum Jun Kim

Axiom Research Group · Department of Applied Mathematics and Computational Science, The Nelson Mandela African Institution of Science and Technology, 404 Nganana, Kikwe, Arumeru, P.O. Box 447, Arusha, 23311, Tanzania · African Institute for Mathematical Sciences, Research and Innovation Centre, KN 3 Road, Gasharu Cell, Kicukiro Sector, Kicukiro District, Kigali, Rwanda +1

PDE Operator LearningPartial Differential EquationsNeural Operators
AskPDF
Aug 14, 2026

From Fixed Grids to Moving Particles:A Transferable Latent Operator for Fluid Dynamics

Lagrangian modeling is vital to fluid dynamics, as it characterizes particle transport and complements the Eulerian representation. However, Lagrangian trajectories are less commonly available than Eulerian fields, while most neural operators are trained and evaluated primarily in the Eulerian representation. This mismatch motivates a new learning problem: can a model trained solely on Eulerian observations generalize zero-shot from Eulerian field prediction to Lagrangian particle rollout, without Lagrangian supervision or task-specific adaptation? To address this problem, we propose the Transferable Latent Operator (TLO), which learns a unified flow representation shared by Eulerian field prediction and Lagrangian particle rollout. TLO decouples latent flow evolution from coordinate-dependent decoding: querying the evolving latent representation at fixed spatial coordinates yields Eulerian fields, whereas querying velocities at particle positions and recursively updating these positions enables Lagrangian rollout. Across five fluid-dynamics benchmarks, TLO consistently outperforms existing neural operators in both Eulerian field prediction and zero-shot Lagrangian rollout, with further gains from limited Lagrangian fine-tuning.

Meng Li, Chuqi Chen, Zhengqing Gao +4

Shanghai Artificial Intelligence Laboratory · The Hong Kong University of Science and Technology · University of Michigan +2

Neural OperatorsComputational Fluid DynamicsPhysics-Based SimulationTransfer LearningFlow Map Learning
AskPDF
Aug 11, 2026

Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEs

We study data-driven prediction of coarse-grained dynamics in multiscale PDE systems. Adopting a closure-free operator-learning viewpoint, we apply a linear coarse-graining map and learn a surrogate evolution operator for the resolved field directly from filtered high-fidelity trajectories. Motivated by the Mori-Zwanzig formalism, we propose a spatiotemporal neural operator mapping a resolved history slab on Ω×[−Tin,0]Ω\times[-T_{\mathrm{in}},0] to a resolved future slab on Ω×[0,Tout]Ω\times[0,T_{\mathrm{out}}]. Spatial mixing uses Fourier convolution, while temporal mixing uses a causal kernel operator with position-attention weights on time lags. This causal temporal operator encodes finite-memory effects in the resolved dynamics while preserving the directionality of the history-to-future map. To improve rollout robustness and suppress nonconservative artifacts, we embed a flux-form inductive bias by parameterizing the windowed update in explicit divergence form. We also provide a data-driven guideline for selecting the memory length TinT_{\mathrm{in}} via the decorrelation time of a closure-injection diagnostic computed from filtered trajectories. We validate on the coarse-grained viscous Burgers' equation, the Kuramoto-Sivashinsky equation, and two-dimensional turbulent flows, obtaining stable autoregressive rollouts with improved long-horizon accuracy and statistical fidelity.

Junfeng Chen

Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong, China

PDE Surrogate ModelingPDE Operator LearningNeural OperatorsFourier Neural Operator
AskPDF
Aug 10, 2026

The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics

Operator learning is a rapidly advancing area of computational science. It is particularly well suited to problems where a partial differential equation (PDE) must be solved repeatedly under varying physical configurations. Most existing architectures represent the solution operator in a fixed basis. While this assumption is well aligned with global structures, it is less suitable for phenomena governed by local interactions in physical space. We explore an alternative perspective motivated by the observation that the continuum limit of coupled oscillator systems can describe a broad class of PDEs. Building on this idea, we introduce the Kuramoto Neural Operator (KNO), which represents the solution through the evolution of a latent field of interacting oscillators. Across a diverse collection of PDE benchmarks, KNO achieves strong predictive performance, with improvements over competing approaches. Our experimental evaluation also includes an extensive ablation study that quantifies the contribution of each architectural component incorporated into KNO. Furthermore, we show that the model's prediction error is closely linked to the collective dynamics of the latent oscillators. It varies systematically with their degree of synchronization, providing insights into the underlying mechanisms.

Petr Badolia, Leonid Obukhov, Dmitry Bylinkin +1

Basic Research of Artificial Intelligence Laboratory (BRAIn Lab) · Innopolis University

Dynamical SystemsKuramoto ModelPDE Operator LearningNeural Operators
AskPDF
Aug 10, 2026

MoNo: Multiscale Optimal Transport Neural Operator for Solving PDEs on General Geometries

Transformer-based neural operators have achieved substantial progress in solving Partial Differential Equations (PDEs) by projecting spatial observations into compact latent tokens and learning physical interactions in latent spaces. However, we reveal that existing learnable projection mechanisms cannot ensure stable and balanced assignments from observation points to latent tokens, causing some latent tokens to be over-assigned while others remain underutilized. This limitation further restricts the design of hierarchical architectures, as assignment imbalance is continuously inherited and amplified across latent spaces, eventually causing severe token collapse in deeper spaces. To address these issues, we propose MoNo (Multiscale Optimal Transport Neural Operator), a progressive multiscale neural operator that efficiently solves PDEs on general geometries through stable latent-space construction. At its core is CoTAP (Cross-scale Optimal Transport Assignment and Projection), a novel latent-space construction method that formulates cross-space assignment between adjacent spaces as an entropy-regularized optimal transport problem, thereby constructing balanced bidirectional projections and stable latent spaces. CoTAP also ensures stable information transfer across multiple latent spaces, further enabling multiscale architectures on general geometries, which in turn support more efficient learning of long-range physical interactions. Extensive experiments demonstrate that MoNo outperforms existing state-of-the-art neural operators in both prediction performance and computational efficiency. Code is available at https://github.com/ZijiangY1116/MoNo.

Zijiang Yang, Xiaomeng Wu, Dongmei Fu

School of Automation and Electrical Engineering, University of Science and Technology Beijing · 2Beijing Engineering Research Center of Industrial Spectrum Imaging

Neural PDE SolversPDE Operator LearningNeural OperatorsOptimal TransportTransformer Neural Operator
AskPDF
Aug 9, 2026

ADEx-FNO: A Unified Ambient-Domain Framework for Fourier Neural Operators on Varying Geometries

Fourier neural operators (FNOs) provide efficient nonlocal spectral learning, but varying geometries and independently chosen discretizations remain difficult to accommodate. We introduce the ambient-domain extension Fourier neural operator (ADEx-FNO), a deterministic framework that incorporates geometry without modifying the defining Fourier-operator layers. Each physical domain is embedded in a fixed ambient hypercube and represented by a signed distance function. Inputs and solution fields are deterministically extended to the ambient domain, transferred to a common, potentially nonuniform rectilinear latent grid, processed by the FNO, then interpolated to an independently chosen target discretization and restricted to the physical domain. All geometry-transfer operations lie outside the optimization procedure and require no trainable graph, point-cloud, deformation, or geometry-decoding modules. ADEx-FNO achieves relative l2 errors of 0.32%-0.77% on held-out smooth-domain nonlinear Poisson and advection-reaction-diffusion problems in 2D and 3D, and is also evaluated on unseen nonsmooth geometries. A single ADEx-FNO inference is then used to initialize conventional CFD solvers. For all 29 converged 2D and 3D RANS cases, pseudo-time iterations decrease, with mean reductions of 44.17% and 43.03%, respectively, with comparable gains across three mesh resolutions. URANS cases reduce post-window physical-time advances by 18.52%-27.51%. In transfer from 2D URANS training data to DNS at different Mach and Reynolds numbers, the bootstrap interval decreases by 23.47%-48.21%, depending on the target statistic. In all CFD tests, ADEx-FNO provides only the initial field; the governing-equation solver controls the subsequent solution, while physical or statistical consistency is assessed separately from computational savings.

Roberto Nuca, Giovanni Testa, Luca Galimberti +1

Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia · Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, Milan, Italy · Physical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

Fourier Neural OperatorPDE Surrogate ModelingPDE Operator LearningNeural Operators
AskPDF
Aug 7, 2026

Neural Operators for Immersed-Boundary Soft Swimmers Locomotion

High-fidelity immersed-boundary simulation resolves the coupled motion of a deforming swimmer and its surrounding flow, but the resulting cost limits repeated evaluations for engineering design, parameter studies, and control. We develop neural-operator surrogates for temporal prediction of the hydrodynamic fields generated by planar and volumetric eel swimmers. The surrogates are trained on regular-grid fields exported from adaptive fluid--structure simulations and are conditioned on swimmer geometry and Reynolds number. The planar model jointly predicts two velocity components, scalar vorticity, and pressure. On five held-out high-Reynolds-number trajectories, its full-domain global relative L^2 error is 3.51 %. The volumetric formulation uses three target-specific models with a common multichannel input: one model predicts three-dimensional velocity, one predicts vorticity, and one predicts pressure. Their full-domain global relative L^2 errors on five held-out within-range trajectories are 3.44 %, 5.58 %, and 19.2 %. Together, the results demonstrate the feasibility of field-resolved neural surrogates for moving-boundary swimmer flows while identifying pressure accuracy and physical consistency as priorities for further development.

Mohammad Sadegh Eshaghi, Yizheng Wang, Navid Valizadeh +2

Chair of Computational Science and Simulation Technology, Institute of Photonics, Department of Mathematics and Physics, Leibniz University Hannover, Germany · Department of Engineering Mechanics, Tsinghua University, Beijing, China · Institute of Structural Mechanics, Bauhaus-Universität Weimar, Germany

PDE Surrogate ModelingRobot LocomotionFluid-Structure InteractionPhysics-Based SimulationNeural OperatorsComputational Fluid Dynamics
AskPDF
Aug 7, 2026

From Points to Edges: Edge-Conditioned Spectral Operators for Physics-Sensitive PDE Learning

Neural operators have become a central tool for solving partial differential equations (PDEs), with spectral operators offering efficient global mixing across spatial locations. However, many PDEs contain physics-sensitive local structures that are critical to the underlying physical behavior. For example, in Darcy flow, local material interfaces are often reflected by sharp changes in the permeability field and can strongly influence the solution. Existing spectral operators primarily adapt modal mixing based on center-point representations, making them insufficiently responsive to such localized structural variations. We propose the Edge-Conditioned Spectral Operator (ESO), a novel spectral operator framework that modulates global spectral mixing using local edge-wise variations. By incorporating the Pairwise-Variation Modal Mixer (PVMM) to inject local edge information into spectral mode selection, ESO preserves the global approximation capability of spectral neural operators while enabling the learned kernel to adapt to physics-sensitive local structures. Furthermore, we introduce a task-adaptive Physics-Aware Reweighting (PAR) that emphasizes physically important regions, identified by taskspecific physical quantities. Across nine PDE benchmarks, ESO consistently achieves state-of-the-art performance. Visual and region-wise analyses further demonstrate that ESO reduces solution errors near coefficient jumps, high-gradient flow structures, and other physically sensitive regions. The code is available at https://github.com/Tanpig-X/ESO.

Zhentao Tan, Ruijie Quan, Yi Yang

Collaborative Innovation Center of Artificial Intelligence (CCAI), Zhejiang University, China

PDE Surrogate ModelingPDE Operator LearningSpectral MethodsNeural Operators
AskPDF
Aug 6, 2026

A neural operator view on U-Nets for inverse imaging problems

Deep neural networks have shown great empirical success in the solution of a wide variety of ill-posed inverse problems in imaging. Yet, very few works have studied their behavior in the limit that turns the discretized ill-conditioned problems into truly ill-posed ones, i.e., for an increasing resolution of the discretization. In this work, we review common approaches to neural operator learning in architectures that resemble a U-Net, one of the most common classical architectures for inverse imaging problems. We discuss advantages and drawbacks of the respective approaches, consider a 1D toy example for improved interpretability, and present extensive numerical experiments on how different types of neural operator U-Nets can improve a first (crude) limited angle CT-reconstruction. In particular, we study how well networks trained for a certain resolution of the discretization generalize to other resolutions. Our finding is that while U-shaped neural operator architectures are by design resolution-invariant, the classical U-Net architecture seems to be more robust with respect to resolution changes than expected.

Alexander Auras, Martin Burger, Samira Kabri +2

Computer Vision Group, University of Siegen, Hölderlinstraße 3, 57076 Siegen, Germany · Helmholtz Imaging, Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, Hamburg, 22607, Germany · Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, Hamburg, 20146, Germany

U-NetNeural OperatorsImage Inverse Problems
AskPDF
Aug 3, 2026

Convex Neural Energy Elements: Monolithic Finite-Element Assembly of Geometry-Parameterized Neural Operators with Stability and Error Guarantees

Extending the neural-operator element method from individually trained, fixed-geometry neural elements to a library of reusable, geometry-parameterized element types fails structurally: a field-predicting operator trained by value regression induces an energy whose assembled Hessian is indefinite, and Newton converges to spurious minima (247% error) even with 1%-accurate field predictions. We introduce convex neural energy elements: each element exports a scalar energy E(g,U), architecturally convex in its boundary degrees of freedom U and smoothly parameterized by its geometry g, realized as a hypernetwork-generated positive-semidefinite quadratic form (an input-convex correction is reserved for non-quadratic physics). A regularization-nullspace principle--the regularizer's nullspace must contain the physics nullspace--removes an otherwise irreducible bias, and assembled elements inherit the classical guarantee that singular element stiffnesses yield a positive-definite global system. We prove conditional error bounds (energy-to-solution accuracy, element-count scaling, geometry generalization) and verify each experimentally. On heat conduction with elliptic holes, one trained element assembles into 2x2 to 8x8 grids and an L-shaped layout of unseen geometries at 0.6-1.0% relative L2 error, with 175x faster per-geometry setup for boundary-quantity workloads. A second trained element type mixes freely with the first in one monolithic assembly, and a three-dimensional instantiation reaches 0.23% on eight-element assemblies--the guarantees are type- and dimension-agnostic. A plane-strain elasticity element, whose physics nullspace is three-dimensional, lands on the analytically predicted regularization floors. Making the energy the learned object turns neural operators from single-use surrogates into reusable elements that inherit the assembly guarantees of the method they extend.

Hongyue Jiang, Jianjiang Zhan, Chenzhuo Zhang +1

School of Civil and Hydraulic Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China · National Center of Technology Innovation for Digital Construction, Huazhong University of Science and Technology, Wuhan, 430074, China

Neural PDE SolversPDE Operator LearningFinite Element MethodNeural Operators
AskPDF