Neural Operators

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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

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

Jun 20, 2026

Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solvers

Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction. We show that this premise is unreliable. An operator trained to the relative L2L^2 error O(10−3)O(10^{-3}) can still produce an initial state in which the discrete Jacobian is indefinite, because the mean-squared training controls error on average while leaving localized pointwise violations of the underlying physics. For a nearly incompressible hyperelasticity problem, we trace this to the predicted volume change: the operator disperses detF\mathrm{det} F well away from one, and the resulting Jacobian acquires negative eigenvalues even when the predicted field is visually indistinguishable from the reference. At a small scale, this is a nuisance; at a multi-million degree-of-freedom scale, it is disqualifying, since the conjugate gradient and other Krylov solvers needed for memory-feasible Newton steps assume a definite spectrum. We then show that a short, label-free fine-tuning phase -- penalizing the operator against the discrete energy, with no additional solution data -- shifts the Jacobian spectrum back to positive definite. Combined with an inexact outer loop, this gives a warm-started Newton method that converges across the full loading range where the unregularized operator fails, reaching up to 5.4×\times wall-clock speedup over incremental continuation on a 3D problem with 6.4 million degrees of freedom.

Jaemin Oh, Youngkyu Lee, Jerome Darbon +1

Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

Spectral RegularizationPDE SolvingHyperelasticityNeural Operators
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Jun 19, 2026

TF-SNO: Time-Frequency Gated Spectral Neural Operators for Learning Non-Stationary Partial Differential Equations

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

Yitian Zhou, Chaoning Zhang, Zhenzhen Huang +8

University of Electronic Science and Technology of China Chengdu, China · Institute of Software, Chinese Academy of Sciences Beijing, China · Harbin Institute of Technology Shenzhen, China

PDE SolvingPDE Operator LearningSpectral MethodsNeural Operators
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Jun 18, 2026

ELADO: Elliptic PDE Assessment Datasets for Operator Learning

We introduce ELADO (Elliptic PDE Assessment Datasets for Operator Learning), a systematic benchmark suite constructed to show and quantify failure modes of neural operator architectures when learning solution operators of elliptic PDEs. While the benchmarks of existing datasets focus on average case performance, the ELADO datasets are constructed to highlight challenges that arise naturally in elliptic PDE problems. In particular, we construct several datasets built around Poisson's equation and the Helmholtz equation, each with non-constant coefficients. We define a controllable data-generating process to create datasets, that are designed to isolate a distinct source of difficulty. Specifically, these are (1) heavy-tailed solution distributions arising from light-tailed coefficient field distributions, (2) spectral distribution shift of the input data, (3) heavy-tailed distributions in the frequency domain of solutions, arising from light-tailed coefficient field distributions, (4) input sensitivity of learned operators, quantified by an empirical local Lipschitz analysis, and (5) the effect of input signal complexity on prediction accuracy under controlled amplitude normalization. We evaluate several neural operator architectures across all datasets and show that heavy-tailed targets, spectral shift, and input sensitivity each cause substantial degradation of the prediction accuracy that standard datasets and metrics (e.g., the mean relative L2L^2 error) may obscure.

Frank Ehebrecht, Toni Scharle, Martin Atzmueller

Osnabrück University, Semantic Information Systems Group, Osnabrück, Germany · ROSEN Technology and Research Center GmbH, Lingen, Germany · German Research Center for Artificial Intelligence (DFKI), Osnabrück, Germany

Synthetic Benchmark GenerationPDE Operator LearningNeural Operators
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Jun 18, 2026

Comparative Study of Neural Surrogate Architectures for Autoregressive Prediction of Internal Battery States

The Doyle-Fuller-Newman (DFN) model resolves internal electrochemical states in lithium-ion batteries with high fidelity. However, the numerical solution of its governing equations is computationally prohibitive for real-time deployment, limiting scalability from individual cells to pack and fleet-scale applications. While machine learning surrogates can substantially reduce inference latency through GPU acceleration, most existing approaches learn solution approximations tied to specific operating conditions rather than learning generalizable state-evolution dynamics. This work presents a systematic comparison of four neural network architectures (MLP, ResNet, U-Net, FNO) formulated as autoregressive state-transition operators that predict full DFN internal states across a wide range of operating conditions. To ensure a controlled architectural comparison, all models are trained under a unified framework using multi-step unrolling and current-conditioning, isolating the impact of spatial inductive bias. Results demonstrate that the U-Net's multi-scale feature hierarchy achieves a mean final-step nRMSE of 3% averaged across all internal state variables after 300-step autoregressive rollouts, while providing a 5.38x speed-up over the numerical solver. These findings highlight spatial inductive bias as a critical determinant of surrogate performance, advancing the development of surrogates for internal state observability for next-generation battery management systems and digital twins.

Gihyun Lee, Thorben Menne, Simon Olma +2

Technische Universität Berlin, Berlin, Germany · IAV GmbH, Berlin, Germany

Neural Surrogate ModelingPDE Surrogate ModelingLithium-Ion BatteriesNeural OperatorsAutoregressive Modeling
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Jun 18, 2026

A fast direct solver based neural network for solving PDEs

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

Jashwanth Reddy Kadaru, Vaishnavi Gujjula

Department of Computer Science & Engineering, International Institute of Information Technology Bangalore (IIIT-B), India · Department of Data Science and Artificial Intelligence, International Institute of Information Technology Bangalore (IIIT-B), India

PDE SolvingPDE Operator LearningNeural Operators
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Jun 17, 2026

ITNet: A Learnable Integral Transform That Subsumes Convolution, Attention, and Recurrence

Convolutional networks, recurrent networks, and transformers each encode different inductive biases -- locality, sequential memory, and content-dependent pairwise interaction -- and have remained mathematically distinct since their inception. We show that this fragmentation reflects not a fundamental diversity in how signals should be processed, but rather incomplete views of a single underlying mathematical object: a learnable integral transform. We introduce the Integral Transform Network (ITNet), a unified architecture built around a learnable kernel that depends jointly on positions and features. This kernel is implemented as a small neural network, specifically an MLP, that models pairwise interactions, enabling the model to adapt its behavior from data. We show that convolution, self-attention (including multi-head), and autoregressive recurrence (including LSTM, GRU, S4, and Mamba) arise as special cases under appropriate parameterizations, and that ITNet is a universal approximator of continuous operators. To make this practical, we develop tiled kernel fusion, importance-weighted Monte Carlo integration, and learned low-rank factorization, enabling efficient and scalable computation. A single ITNet architecture with a shared operator and lightweight modality-specific encoders matches or exceeds specialized baselines on ImageNet-1K , GLUE, ModelNet40, VQA,v2 and NLVR2. The results demonstrate that a single learned interaction mechanism can recover the behavior of all three architectural families from data.

Ashim Dhor, Rasel Mondal, Pin-Yu Chen

Indian Institute of Science Education and Research Bhopal · IBM Research

Transformer AttentionConvolutional Neural NetworksNeural Operators
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Jun 16, 2026

Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problems

Operator learning is an emerging field at the intersection of machine learning and scientific computing. By learning mappings between function spaces, neural operators provide data-driven surrogate models for families of partial differential equations (PDEs). Once trained, these models can evaluate solution operators efficiently, making them suitable for many-query applications such as real-time prediction and parameter sweeps. However, maintaining high approximation accuracy and stable long-term predictions remains challenging for complex forward and inverse problems. To address these challenges, we propose the Starter-Iterator Neural Operator (SINO), which incorporates the initialization and residual-correction structures of classical iterative solvers into neural operator learning. The frequency-domain Starter captures dominant global spectral features and provides an informed initial approximation, while the latent-space Iterator applies successive residual-based corrections to refine local and multiscale solution structures. Experiments on representative time-dependent PDEs, including the Navier-Stokes and acoustic wave equations, together with applications to image super-resolution and weather forecasting, show that SINO achieves competitive accuracy and stable performance across the benchmarks considered in this work.

Kuilin Qin, Lianfang Wang, Xu Sun +4

PDE Surrogate ModelingPDE Operator LearningPDE Inverse ProblemsNeural Operators
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Jun 16, 2026

Geometry-Aware Post-Hoc Uncertainty Quantification in Operator Learning

Neural operators provide fast surrogates for PDEs but their deterministic predictions limit their use in tasks requiring uncertainty quantification (UQ), especially under geometric variability. Existing approaches primarily model uncertainty in network parameters, largely overlooking the geometry-aware representations learned by the operator itself. We propose REEF-GP (Residual on Embedded Features Gaussian Process), a post-hoc UQ framework that fits a GP to the residuals of a frozen neural operator whose internal embeddings define the kernel feature space. Rather than learning a separate feature map, REEF-GP adapts the operator's intrinsic coordinate-feature representations to construct geometry-aware uncertainties. To ensure stability and scalability on unstructured domains, REEF-GP incorporates spectral-normalized projections, heteroscedastic geometry-aware noise, and efficient subset-based training that avoids restrictive low-rank approximations. Across five PDE benchmarks with varying geometries, REEF-GP preserves predictive accuracy while achieving calibrated uncertainty estimates competitive with deep ensembles but at a fraction of their cost. Our approach remains robust under geometric distribution shift, with uncertainty concentrating in physically meaningful regions (e.g., shock fronts). Our results demonstrate that accurate and scalable post-hoc UQ for neural operators can be achieved directly in their learned feature space, offering a practical alternative to parameter-centric approaches.

Oriol Vendrell-Gallart, Nima Negarandeh, Ramin Bostanabad

Department of Mechanical and Aerospace Engineering University of California, Irvine Irvine, CA 92617

Uncertainty QuantificationPDE Surrogate ModelingNeural OperatorsGaussian Process Regression
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Jun 16, 2026

Operator Boosting Produces Pareto-Efficient PDE Surrogates

Neural operators are widely used as surrogate solution maps for partial differential equations (PDEs), but full-size models can be costly to store, deploy, and evaluate in many-query scientific workflows. This work introduces Operator Boosting, a stagewise residual-learning framework for constructing compact neural-operator surrogates directly, rather than training a large model and compressing it afterward. Starting from the empirical mean predictor in normalized output coordinates, the method trains a sequence of tiny same-family neural operators on residual fields and incorporates each correction through validation-selected shrinkage. We instantiate the framework with Fourier neural operators (FNOs), DeepONets, and convolutional neural operators (CNOs), and compare boosted tiny stacks against full-size monolithic baselines across one-, two-, and three-dimensional PDE benchmarks from PDEBench, APEBench, and The Well. Across 30 dataset-architecture pairs, 21 show positive mean accuracy gains and 17 have positive confidence intervals, while all boosted stacks reduce trainable parameter count by approximately 72-95%. Best-model comparisons show empirical Pareto improvements on 7 of 10 completed PDE benchmarks, including two-dimensional Navier-Stokes, shallow-water dynamics, Darcy flow, one-dimensional transport and reaction systems, and three-dimensional compressible Navier-Stokes. These results show that Operator Boosting often improves the empirical accuracy-parameter Pareto frontier of neural PDE surrogates, while also exposing PDE- and architecture-dependent regimes where residual boosting fails to offset compression.

Lennon J. Shikhman

Georgia Institute of Technology College of Computing Atlanta, Georgia, USA

PDE Surrogate ModelingDeep Operator NetworksNeural Network CompressionNeural Operators
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Jun 16, 2026

Generalization Guarantees for Multi-Input Neural Operator Learning in Sobolev Spaces

We develop approximation and generalization error estimates for multi-input neural operators, with the output error measured in Sobolev norms. In contrast to standard operator-learning settings with a single input function, our framework allows multiple input functions defined on possibly different domains, with different dimensions and Sobolev regularities. The derived rates explicitly quantify the contribution of each input space to the final error bound. In particular, in the balanced regime, the approximation and generalization rates are governed by the interaction between the input dimensions, regularities, and Sobolev orders, while the dependence on the model complexity retains a log⁡log⁡/log⁡\log\log/\log-type structure. Our analysis provides a general theoretical framework for multi-input operator learning, including Sobolev training, and is applicable to operator learning problems arising from partial differential equations and scientific computing.

Yahong Yang, Zecheng Zhang, Wei Zhu +2

School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA. · Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA. · Department of Mathematics, Hong Kong Baptist University, FSC1202, Fong Shu Chuen Building, Hong Kong Baptist University, Kowloon Tong, Hong Kong.

Neural Network Approximation TheoryOperator LearningNeural OperatorsGeneralization Bounds
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Jun 15, 2026

Factorized Neural Operators Decompose Dynamic and Persistent Responses

Physical systems often exhibit heterogeneous mechanisms, where rapidly evolving dynamics coexist with persistent structures. Capturing such multiscale physical behavior remains challenging for existing neural operators, which typically rely on single dominant inductive bias and therefore couple distinct physical responses into a shared representation. We introduce the Unified Green's Function Framework across domains and propose the Factorized Neural Operators (FaNO), which decompose spectral representations into equivariant dynamic responses and invariant persistent responses, leading to better interpretability and generalization. Mechanistically, we show that the two operator branches spontaneously specialize into distinct physical roles that remain consistent across scales and domains: the equivariant branch captures rapidly varying transient dynamics, whereas the invariant branch extracts coherent persistent structures. This factorized mechanism of FaNO improves prediction accuracy, parameter efficiency and cross-scale generalization across physical systems and domains. In particular, it maintains consistent predictions under long-horizon autoregressive rollout, cross-resolution extrapolation and physical-regime shifts. These findings suggest that scalable physical modeling may benefit from moving beyond single-inductive-bias formulations toward factorized operator representations that better reflect the heterogeneous organization of physical systems, accelerating the reliable deployment of machine learning for scientific computing and discovery.

Hao Tang, Yuechen Duan, Jiongyu Zhu +3

School of Medicine, University of Dundee, United Kingdom. · School of Data Science, Fudan University, China. · School of Mathematical Sciences, Fudan University, China. +3

Mechanistic InterpretabilityPDE Operator LearningInvariant Representation LearningNeural Operators
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Jun 15, 2026

PhysGuard: Fisher-Guided Gradient Projection for Sim-to-Real Neural PDE Surrogates

Neural operator models trained on simulation data often lose accuracy when applied to experimental measurements due to the sim-to-real gap. Standard fine-tuning with limited real data can reduce this gap, but it may also damage the core physics-relevant representations learned during pretraining. Although knowledge-preserving adaptation has been widely investigated in vision or language tasks, it remains unclear whether these methods are suitable for neural operators whose architectures and protected knowledge are fundamentally different. Neural operators need to preserve core-scale physical structures rather than semantic or visual features. We propose PhysGuard, a physics-preserving framework for accurate sim-to-real adaptation of neural operators. Specifically, PhysGuard uses the empirical Fisher Information Matrix computed on simulation data to identify physics-critical parameter directions, then restricts fine-tuning updates to directions that do not interfere with them. A layer-wise Gram-matrix formulation makes this efficient for models with millions of parameters, while an adaptive threshold automatically determines the protected subspace size. A spectral probe experiment shows that the dominant Fisher directions are strongly associated with low-frequency output structures. Experiments on benchmark across four neural operator architectures and different physical systems show that PhysGuard performs strongly on most evaluation metrics compared to baselines. The benefits are most evident under severe domain shift, where it reduces low-frequency error by up to 32% compared to standard fine-tuning while maintaining adaptability. Our code is available at https://github.com/Data-Driven-Computational-Geotechnics/PhysGuard.

Changjian Zhou, Junfeng Fang, Negin Yousefpour +3

Faculty of Engineering and IT, University of Melbourne · School of Computing, National University of Singapore · Artificial Intelligence Research Institute, IFLYTEK Co., Ltd.

Synthetic-to-Real Domain AdaptationPDE Surrogate ModelingNeural Operators
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Jun 13, 2026

Towards Data-Efficient Cross-Device Generalization of Grad-Shafranov Equilibria via Transfer Learning Neural Operator

Real-time reconstruction of magnetohydrodynamic equilibria is essential for plasma shaping, stability assessment and feedback control in magnetic confinement fusion. However, Grad-Shafranov equilibrium calculations remain largely device-specific and iterative, limiting their use in latency-constrained control settings. Existing neural approaches can accelerate individual equilibrium predictions, but they do not generally provide reusable models across changing plasma boundaries or tokamak geometries. Here we show that equilibrium reconstruction can be recast as a cross-device operator learning problem. We develop a domain-specific neural operator framework that maps geometry and profile parameters directly to the poloidal flux field, replacing repeated solve-on-demand computation with amortized operator inference. Using the analytically tractable Solov'ev family as a controlled Grad-Shafranov testbed, we generate equilibria across eight geometrically distinct tokamak-like configurations and benchmark five neural operator architectures under four transfer-learning strategies. Single-geometry pretraining gives poor transfer to unseen devices, whereas multi-geometry pretraining enables data-efficient adaptation. The Wavelet Neural Operator gives the strongest cross-geometry performance, reaching mean relative L2 errors below 4% with 100 labelled target equilibria and below 2% with full fine-tuning. The predicted magnetic fields satisfy the divergence-free constraint to numerical precision, and four architectures achieve millisecond or sub-millisecond inference. These results identify neural operator pretraining as a route towards reusable, real-time equilibrium inference across fusion device configurations.

Jay Phil Yoo, William Howes, Yashika Ghai +3

Grainger College of Engineering, Nuclear, Plasma & Radiological Engineering Department, University of Illinois Urbana-Champaign, Urbana, IL, USA · Fusion Energy Division, Oak Ridge National Lab, Oak Ridge, TN, USA · National Center for Supercomputing Applications, Urbana, IL, USA +2

Transfer LearningPDE Operator LearningNeural OperatorsData-Efficient LearningMagnetohydrodynamics
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Jun 12, 2026

Zero-shot generalization of transformer neural operators to larger domains

Transformer-based neural operators have shown remarkable performance for approximating solution operators of partial differential equations on complex geometries. However, existing approaches implicitly assume a fixed domain size, which limits their ability to generalize at inference. In this work, we investigate domain extension, namely zero-shot inference on spatial domains that are significantly larger than those encountered during training. We argue that this setting fundamentally requires spatial locality and translation equivariance. We propose to implement this locality via a decomposable bias in the attention logits computation, enabling finely controllable locality while remaining fully decomposable into query-key inner products and directly compatible with optimized attention kernels. Combined with rotary positional embeddings, it enables expressive embeddings with controllable spatial support without altering the transformer architecture. We empirically show that our approach substantially improves zero-shot generalization to larger domains across two PDE benchmarks and a 3D industrial atmospheric flow application. Our code and datasets are available at https://github.com/cerea-daml/domain-extension.

Armand de Villeroché, Sibo Cheng, Vincent Le Guen +5

CEREA, ENPC, EDF R&D, Institut Polytechnique de Paris, Île-de-France, France · SINCLAIR AI Laboratory, Saclay, Île-de-France, France · EDF R&D, Île-de-France, France +1

Transformer Neural OperatorLocal AttentionPDE Operator LearningAttention MechanismsNeural Operators
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Jun 12, 2026

A Fixed-Point Neural Operator for Size- and Functional-Transferable Hamiltonian Prediction

Predicting the Kohn-Sham Hamiltonian with machine learning can accelerate density functional theory while retaining access to molecular orbitals, energy levels, and electronic-structure observables that energy-only surrogates cannot resolve. Yet element-wise agreement with the converged Hamiltonian, an implicit fixed point of the self-consistent field iteration, does not determine the occupied subspace that governs orbital energies and densities. Here we present HamEvo, a neural operator that learns the single-step self-consistent update and returns the converged Hamiltonian as its fixed point. HamEvo is pre-trained on intermediate self-consistent trajectories and calibrated at equilibrium with density-matrix supervision. Across benchmarks from MD17 to drug-like QMugs, HamEvo lowers Hamiltonian errors by 35-49% over direct-regression and deep-equilibrium baselines, and predicts QMugs HOMO and LUMO energies with mean absolute errors of 0.036 and 0.053 eV, near the 1 kcal/mol chemical-accuracy scale. Few-shot fine-tuning with only 20 reference conformations extends HamEvo to molecules of up to 122 atoms, well beyond the size range covered by pre-training. With thermal molecular-dynamics sampling, HamEvo captures temperature-dependent HOMO-LUMO gap renormalization beyond the harmonic approximation. Inference is up to 242 times faster than conventional DFT.

Yunhong Lou, Xihang Yue, Xinran Wei +2

1Zhejiang University, Hangzhou, China. · 2Zhongguancun Academy, Beijing, China. · 3Zhongguancun Institute of Artificial Intelligence, Beijing, China.

Quantum ChemistryMolecular Property PredictionOperator LearningNeural OperatorsDensity Functional Theory
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Jun 11, 2026

How Much Memory Do We Need? Adaptive Memory Gate for Neural Operators

Neural operators have emerged as a powerful data-driven approach for solving time-dependent PDEs. Among recent advances, memory-augmented neural operators explicitly incorporate past states and have achieved remarkable performance under low-resolution observation settings. However, existing approaches apply a fixed memory weight regardless of observation conditions, such as resolution or physical parameters, limiting their adaptability. Our preliminary experiments reveal that optimal memory weight varies with resolution and viscosity, implying that a fixed memory weight cannot simultaneously optimize performance across diverse settings. We propose AMGFNO, which dynamically modulates memory weight through a learnable gate. On the Kuramoto-Sivashinsky and Burgers' equations, AMGFNO achieves 55-79% nRMSE reduction over at low resolution, with the learned gate value automatically decreasing from gˉ≈0.7\bar{g} \approx 0.7 to near-zero as resolution increases.

Jihyeon Hur, Yongseok Kwon, Min-Gi Jo +2

1KAIST, Republic of Korea.

Memory-Augmented Neural NetworksFourier Neural OperatorPDE Operator LearningNeural Operators
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Jun 10, 2026

Harness In-Context Operator Learning with Chain of Operators

Neural operators approximate mappings between function spaces, but often generalize poorly to other operators and usually require fine-tuning or retraining. In-Context Operator Networks (ICON) addresses this issue by prompting the model with numerical context so that the model learns specific operators from prompts and adapt to different operators without fine-tuning. However, ICON may still fail to generalize to out-of-distribution (OOD) operator tasks. Inpired by the success of harness engineering of Large Language models (LLMs), we introduce Chain of Operators (CHOP), a framework that harness a frozen ICON to OOD operator tasks without updating its parameters. Specifically, CHOP constructs a chain of operators consisting of explicit elementary transformations and the frozen ICON. Experiments on a scalar conservation law and a mean-field control problem show that CHOP reduces relative inference error over direct ICON evaluation, while each operator in the chain remains interpretable and in closed form. A chain constructed on one PDE family further generalizes to a different family, indicating shared mechanisms across harness systems.

Minghui Yang, Ling Guo, Liu Yang

Department of Mathematics, Shanghai Normal University, Shanghai, China · Department of Mathematics, National University of Singapore, Singapore

OOD GeneralizationPDE Operator LearningOperator LearningNeural Operators
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Jun 10, 2026

HAMNO: A Hierarchical Adaptive Multi-scale Neural Operator with Physics-Informed Learning for Dynamical Systems

Neural operators provide a powerful framework for learning solution mappings of partial differential equations directly in function space. However, many existing architectures still struggle to represent nonlinear time-dependent systems that involve multi-scale structures, long-range interactions, and stable long-time evolution. In this work, we introduce the Hierarchical Adaptive Multi-scale Neural Operator (HAMNO), a neural-operator architecture that combines local convolutional representations, global spectral operators, and hierarchical encoder-decoder processing. The central component of HAMNO is a data-dependent gating mechanism that adaptively balances local and global information at each spatial location, allowing the model to resolve fine-scale features while preserving long-range dependencies. We further develop a physics-informed extension, PI-HAMNO, based on a multi-objective loss strategy that combines data fitting with strong- and weak-form physics constraints. The strong-form term penalizes the domain-integrated squared PDE residual in physical coordinates, while the weak-form term is constructed by multiplying the governing residual by finite-element test functions and evaluating the resulting element integrals using centroid-based tetrahedral quadrature. The framework is evaluated on non-periodic Allen-Cahn (AC), Cahn-Hilliard (CH), and Swift-Hohenberg (SH) equations defined on cubic domains. Across long-horizon rollout, data-limited training, out-of-distribution initial-condition shifts, and random-seed variations, HAMNO improves predictive accuracy over standard neural-operator baselines, while PI-HAMNO further enhances stability, physical consistency, and data efficiency. The implementation is publicly available at https://github.com/MBamdad/HAMNO .

Mostafa Bamdad, Mohammad Sadegh Eshaghi, Timon Rabczuk

Chair of Computational Mechanics, Institute of Structural Mechanics, Bauhaus-Universität Weimar, 99423 Weimar, Germany · Chair of Computational Science and Simulation Technology, Institute of Photonics, Department of Mathematics and Physics, Leibniz University Hannover, 30167 Hannover, Germany

Neural PDE SolversPhysics-Informed Neural OperatorsNeural Operators
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Jun 8, 2026

Topological Neural Operators

We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains. TNOs represent data as features defined on cells of varying dimension and model their interactions through Discrete Exterior Calculus, enabling explicit cross-dimensional coupling via gradient-, curl-, and divergence-type operators. The key design principle is to decouple where information flows, as governed by fixed topological operators, from how it is transformed (which is learned), yielding models that respect the geometric support of physical quantities and expose conservation and compatibility structure. We further propose Hierarchical TNOs (HTNOs), which incorporate learned coarse complexes to propagate long-range and topology-dependent information. Our framework subsumes existing NOs as a special case, providing a unified perspective on operator learning across discretizations. Across a range of PDE benchmarks, including irregular-geometry flow problems, TNOs and HTNOs improve accuracy; controlled studies further isolate the benefits of native higher-rank and topological structure. Project page: https://circle-group.github.io/research/TNO

Lennart Bastian, Samuel Leventhal, Mustafa Hajij +1

Imperial College London, UK · University of San Francisco, USA

PDE Operator LearningNeural Operators
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Jun 8, 2026

Geometry-Aware Anisotropic Boundary Correction for Aerodynamic Simulation

Aerodynamic simulation is a key component of engineering shape design, where core quantities such as the surface pressure coefficient strongly depend on flow dynamics near solid boundaries. Neural operators provide an efficient alternative to expensive Computational Fluid Dynamics (CFD) solvers. However, conventional methods treat the boundary region isotropically, failing to account for the distinct physical behaviors along the boundaries. In reality, the aerodynamic process exhibits anisotropy: along the tangential direction, flow propagates along the wall; along the normal direction, physical quantities are constrained by the wall. To explicitly model the distinct physical behaviors, we propose GeoABC, a geometry-conditioned anisotropic boundary correction framework. GeoABC leverages the boundary geometries to introduce direction-aware boundary correction into the intermediate representations of neural operators, transforming boundary geometry from static input features into a structural prior that modulates physical prediction. On 2D airfoil and 3D car tasks, GeoABC consistently adapts to multiple neural operator backbones, reducing near-boundary relative L2L_2 error by ∼\sim38% on average, narrowing the structural near-wall gap shared by mainstream neural operators, and advancing neural operators toward high-fidelity aerodynamic simulation.

Xin Zhang, Yipeng Huang, Shu Jiang +2

School of Informatics, Xiamen University, Xiamen, China · Institute of Artificial Intelligence, Xiamen University, Xiamen, China

Neural OperatorsComputational Fluid Dynamics
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Jun 8, 2026

From inverse problems to neural operators: prediction, mechanism, and generalization of data-driven models

Scientists have historically relied on mathematical models based on differential equations to relate system inputs -- forces, fluxes, or heat sources -- to outputs, such as displacement, velocity, concentration, and temperature. These models rely on deep domain knowledge to determine the form of the governing differential equation, which is then calibrated with data by solving an inverse problem. In recent years, the field of Scientific Machine Learning has introduced a variety of alternative modeling strategies for physical systems. A method called Sparse Identification of Nonlinear Dynamics learns the governing equation as a sparse linear combination of terms in a user-defined library. Neural Ordinary Differential Equations construct the governing equation by taking in the state and its derivatives at the input layer of a neural network. Entirely foregoing the modeling framework of differential equations, neural operators directly learn a non-linear mapping between the system inputs and outputs. From inverse problems to neural operators, all of these modeling strategies can be conceptualized as data-driven machinery to predict a system's response over a range of inputs. It is then natural to wonder how exactly these various strategies relate to each other, and whether they can be neatly taxonomized. Drawing from the philosophical literature on scientific models, we argue that many model types have a common structure, differing only in the assumed model class of the input-output relation they define. Connecting to philosophical ideas on mechanism, and arguing that data from physical systems arises from solutions to parsimonious differential equations, we propose that only certain models are capable of mechanism discovery, and thus generalization. Our analysis is intended to unite apparently disparate modeling strategies and provide insight into their appropriate use cases.

Conor Rowan

Aerospace Engineering2026 University of Colorado Boulder 3775 Discovery DriveJun 9 Boulder, CO 80309

Scientific DiscoveryScientific MLNeural OperatorsInverse Problems
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Jun 7, 2026

Conformal Prediction for Neural Operators: Distribution-Free Uncertainty Quantification in Physics Simulation

Neural operators such as the Fourier Neural Operator (FNO) have emerged as powerful surrogates for solving partial differential equations (PDEs), achieving speedups of several orders of magnitude over traditional numerical solvers. However, deploying these models in safety-critical engineering applications -- such as thermal management of electronic components and battery systems -- requires not only accurate point predictions but also rigorous uncertainty guarantees. Existing uncertainty quantification (UQ) methods for neural operators, including Monte Carlo Dropout and Deep Ensembles, provide only relative uncertainty estimates without formal coverage guarantees. In this work, we propose the first application of split conformal prediction to neural operator-based physics simulation, providing distribution-free prediction intervals with finite-sample coverage guarantees. We further introduce a normalized conformal prediction scheme that leverages MC Dropout uncertainty to produce adaptive-width intervals, yielding tighter intervals in regions of low uncertainty and wider intervals where the model is less certain. Full-scale experiments (33.7M parameters, 800 training samples, 5 ensemble members, NVIDIA V100) on steady-state heat conduction benchmarks demonstrate that our method achieves 89.1% empirical coverage at the target level of alpha=0.1, while producing spatially adaptive prediction intervals that reflect the underlying physical uncertainty structure. We also provide an uncertainty decomposition framework that separates epistemic uncertainty (68% of total) from aleatoric uncertainty (32% of total), offering actionable guidance for data collection and model improvement. Our method is implemented in an open-source platform with REST API endpoints and interactive 3D visualization.

Michael Chin

Independent Researcher

Uncertainty QuantificationPDE Surrogate ModelingConformal PredictionNeural OperatorsRegression Prediction Intervals
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Jun 6, 2026

GENERIC-FNO: Embedding Energy Conservation and Entropy Production into Fourier Neural Operators

We propose GENERIC-FNO, a neural operator that embeds the metriplectic (GENERIC) degeneracy structure of nonequilibrium thermodynamics in function space, coupling reversible, energy-conserving dynamics to irreversible, entropy-producing dynamics via the degeneracy conditions. Prior structure-preserving neural operators enforce at most one conservation law or a Hamiltonian form, and thermodynamically consistent learning has been confined to finite-dimensional, graph, or particle systems. GENERIC-FNO learns the energy and entropy functionals as neural operators and builds the reversible and irreversible operators as diagonal Fourier multipliers flanked by rank-one projections that enforce both degeneracy conditions exactly, by construction, with no penalty, update projection, or residual; the Jacobi identity is not enforced. The identities hold to machine precision (~10^-13) for any initialization, dimension, or resolution, so the continuous-time dynamics conserve the learned energy and produce the learned entropy exactly, with explicit time stepping adding only an O(dt^2) drift. These are guarantees about the learned functionals within GENERIC's scope of closed conservative-dissipative dynamics, not a certificate of physical accuracy, and the (E,S,L,M) decomposition is not unique; we make this gauge freedom explicit and propose a gauge-invariant dissipation diagnostic independent of the learned functionals. Across three backbones (1D/2D FNO, DeepONet) and four canonical scalar PDEs, the guarantees transfer zero-shot over a 4x super-resolution range and hold in 3D; the diagnostic identifies the reversible and the most dissipative system in every backbone; and over 200-step rollouts, where every unconstrained model we test diverges or collapses, GENERIC-FNO stays bounded, at half the parameters but 4-10x the compute, while losing accuracy on pure transport and on the smallest 1D backbone.

Jason Sulskis, Sathya Ravi

Department of Computer Science, University of Illinois at Chicago · Applied Systems Laboratory, Systems Integration Division, Georgia Tech Research Institute

Fourier Neural OperatorPhysics-Informed Neural OperatorsNeural Operators
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Jun 2, 2026

Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations

Neural operator surrogates (NO) approximate PDE solutions orders of magnitude faster than numerical solvers, but suffer from spectral bias: high-frequency content is systematically attenuated, limiting reliability where fine-scale structure matters. Sparse sensor measurements of the field are often available too, offering pointwise accuracy without spectral distortion but covering only a small fraction of the domain. We address this by treating NO predictions as auxiliary observations in a diffusion posterior sampling framework. Our method, FreqNO-DPS (https://github.com/niccoloperrone/FreqNO-DPS), combines an unconditional score-based diffusion prior, trained on high-fidelity simulations, with diffusion posterior sampling (DPS) conditioned on sparse observations and guided by a frozen neural operator. Naive integration reintroduces the surrogate's spectral bias; we resolve this with a closed-form, spectrally shaped guidance score that weights the surrogate by its frequency-dependent accuracy and needs no denoiser backpropagation. A distribution-free analysis bounds the approximation error across the frequency-diffusion-time plane and shows the guidance's frequency dependence is preserved regardless of distributional assumptions. On 3D elastic wavefield prediction at 5% and 2% sensor coverage, the method reaches near-zero spectral bias across all bands, where both the surrogate and sensor-only DPS show systematic high-frequency attenuation. Isotropic guidance, the natural baseline, improves pointwise accuracy but carries the bias into the posterior nearly intact, confirming that frequency-dependent calibration is essential, not merely beneficial. The framework needs only paired surrogate/reference data and exploits no problem-specific structure beyond the residual's approximate spectral diagonality, verifiable for new surrogates via the coherence diagnostic we provide.

Niccolò Perrone, Fanny Lehmann, Stefania Fresca +1

Université Paris-Saclay, CentraleSupélec, CNRS, ENS Paris-Saclay · Laboratoire de Mécanique Paris-Saclay UMR 9026 · 8-10 rue Joliot Curie, 91190 Gif-sur-Yvette, France +5

PDE Surrogate ModelingSpectral BiasDiffusion-Based Inverse ProblemsDiffusion Posterior SamplingNeural Operators
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Jun 2, 2026

Let There Be Light: Reflection, Refraction and Scattering for Neural Operators

Neural operators learn mappings between infinite-dimensional function spaces and provide a data-driven surrogate modeling paradigm for parametric partial differential equations (PDEs). Existing architectures typically obtain expressivity by parameterizing integral kernels in prescribed transform domains or by applying attention-like interactions over discretized spatial points. While these approaches have achieved substantial progress, they often face a persistent trade-off among physical interpretability, nonlocal spatial communication, mesh scalability, and computational cost. We propose a Light-inspired neural operator(LiNO), an operator-learning architecture whose latent evolution is decomposed into three mechanisms motivated by elementary light transport: reflection, refraction, and scattering. Reflection and refraction act as adaptive pointwise transformations in latent feature space, enabling local feature reorientation and anisotropic modulation, whereas scattering performs input-dependent nonlocal propagation over the physical domain. We first formulate scattering as a normalized pairwise kernel with relative positional bias, and then develop an efficient scattering variant that replaces explicit pairwise interactions with positive-feature global propagation and a local diffusion branch, reducing the dominant spatial complexity from quadratic to linear. This yields a structured neural operator that separates local feature modulation from global spatial communication while retaining a modular and interpretable latent evolution.

Keke Wu, Yixuan Zhang, Jingrun Chen

Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou 215123, China · School of Artificial Intelligence and Data Science, University of Science and Technology of China, Hefei 230026, China · Suzhou Big Data & AI Research and Engineering Center, Suzhou 215123, China +2

PDE Operator LearningNeural Operators
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Jun 2, 2026

EqGINO: Equivariant Geometry-Informed Fourier Neural Operators for 3D PDEs

Deep learning surrogates for 3D Partial Differential Equations (PDEs) often fail to generalize across geometric transformations because they depend heavily on specific coordinate systems. While equivariant networks offer a solution, they typically rely on local operations in the spatial domain, making the global receptive field, which is essential for PDE dynamics, computationally expensive. Conversely, Fourier Neural Operators (FNOs) efficiently capture global interactions, yet establishing 3D equivariance within them remains impractical due to the prohibitive cost of spectral group convolutions. To bridge this gap, we introduce EqGINO, a geometrically robust framework that enforces isotropy in the spectral domain. By design, EqGINO guarantees exact equivariance to the discrete symmetries inherent to the discretized computational domain. Beyond this discrete guarantee, our structural prior enables effective generalization to arbitrary continuous orientations even with a limited number of SE(3)-transformed training samples. Consequently, our method robustly models coordinate-invariant physical laws on complex irregular 3D geometries. Our code is available at https://github.com/sung-won-kim/EqGINO

Sungwon Kim, Juho Song, Seungmin Shin +3

Graduate School of Data Science, KAIST, Daejeon, Republic of Korea · Mathematical Sciences, KAIST, Daejeon, Republic of Korea · LG Electronics, Pyeong-taek, Republic of Korea +1

Fourier Neural OperatorPDE Surrogate ModelingPDE Operator LearningNeural Operators
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Jun 2, 2026

Will Accurate Fields Mislead Photonic Design? FromGlobal Accuracy to Port Readout

Neural field surrogates can accelerate photonic design loops, but a surrogate that looks accurate in global field error can still mis-rank candidate devices when the final decision depends on localized output-port readouts. This risk is acute in propagation-dominated MMI splitters and couplers, where port power, splitting, phase, and coupling are determined by accumulated modal interference and output-window aggregation rather than by average field similarity alone. We study this field-to-design mismatch through a Field/Mediator/Readout view that separates dense complex-field error from propagation-profile and output-window errors before port aggregation. To align the surrogate with this chain, we propose PaNO, a propagation-aligned neural operator that keeps the full-field prediction interface while organizing latent states around local boundary structure, transverse modal content, axial propagation, and cross-mode interaction. We also evaluate PaNO-R2, an output-aware feedback variant for residual field components near the port region. On a 15-wavelength tunable 3×33{\times}3 MMI benchmark with 4608 held-out fields, PaNO lowers NeurOLight's port-power error from 0.2018 to 0.0739 despite slightly higher cMAE, showing that global field accuracy alone is not sufficient for design-relevant readout fidelity. PaNO-R2 attains the best cMAE, propagation-profile error, output-profile error, and port-power error, reducing NeurOLight's port-power and output-profile errors by 72.7% and 72.5%.

Yitian Zhang, Yonghong chen, Youming Chen +6

Sun Yat-sen University · The Hong Kong University of Science and Technology (Guangzhou)

Neural Surrogate ModelingPhysics-Based SimulationNeural Operators
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Jun 1, 2026

Spectral Audit of In-Context Operator Networks

Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit for in-context operator learning. For a fixed prompt, we differentiate the network output with respect to the query function and view the resulting Jacobian as a learned tangent operator. Projecting it onto Fourier modes, we obtain a local spectral characterization of the inferred operator, including frequency-dependent gains, phase structure, and cross-mode coupling. The audit complements standard prediction metrics by testing whether the model reproduces local mechanisms of the underlying PDE operator rather than only outputs. Across benchmarks, the audit reveals distinct operator-level phenomena, including phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction--diffusion stability structure. It also detects failures partially hidden by prediction-error metrics, including high-frequency degradation, incorrect phase recovery, and prompt--operator inconsistencies. Corrupted or internally inconsistent prompts lead to degraded tangent-operator structure even when pointwise predictions remain partially accurate. Our results suggest that prediction accuracy and local operator fidelity are distinct properties of learned neural operators. Our framework also provides a diagnostic for stability, sensitivity, and operator consistency.

Zhiwei Gao, Liu Yang, George Em Karniadakis

Division of Applied Mathematics, Brown University, Providence, RI, USA · Department of Mathematics, National University of Singapore, Singapore

PDE Operator LearningOperator LearningNeural Operators
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May 31, 2026

Variation Spaces for Encoder--Decoder Neural Operators: Approximation and Generalization

Inspired by the function-space theory of neural networks, we formulate and analyze a variation space for nonlinear operators between Hilbert spaces, defined through vector-valued Borel measures of bounded variation. We characterize its unit ball as the closed convex hull of a vector-valued single-neuron dictionary in Bochner spaces. For the ReLU activation, the bounded linear operators in this space are precisely the Schatten-11 operators, with equivalent norms. For operators in this space, we establish encoder--decoder approximation bounds in the Bochner LqL^q-norm, where the error decomposes into input and output encoding errors and a finite-width term of order N−1/2N^{-1/2}. Under sub-Gaussian assumptions on the input and noise, we further derive high-probability generalization bounds for empirical least squares over path-norm-constrained encoder--decoder networks; the finite-sample contribution to the squared prediction error is of order K−1/2K^{-1/2} up to logarithmic factors. The finite-width and finite-sample constants are independent of the encoding dimensions and bases, with the latter also independent of the network width. When the encoding errors decay algebraically, these bounds yield algebraic approximation and learning rates, in contrast to the complexity barriers for Lipschitz and Fréchet differentiable operator classes.

Jia-Qi Yang, Lei Shi

School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China.

Neural Network Approximation TheoryOperator LearningNeural Operators
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May 31, 2026

Revisiting Neural Processes via Fourier Transform and Volterra Series

Modeling unknown latent functions from finite, irregularly sampled measurements is a recurring challenge across science and engineering. Neural processes (NPs), a family of probabilistic functional models, are promising solutions -- especially when endowed with domain-specific symmetries like translation equivariance, which improve sample efficiency and generalization. Yet existing translation-equivariant NPs face two limitations: (i) they stack generic components with non-linearities, obscuring the induced function class and limiting interpretability; and (ii) convolutional designs are limited by local receptive fields and the need to embed inputs onto a dense uniform grid, while attention-based alternatives lift these restrictions at quadratic cost in the number of observations. We address both with two contributions. First, using the Volterra expansion, we approximate continuous translation-equivariant operators by sums of higher-order convolutions, yielding analytical transparency while admitting efficient evaluation via first-order convolutions. Second, we introduce set Fourier convolutions (SFConvs), a frequency-domain parameterization that operates directly on irregularly sampled points, achieves approximately global receptive fields, and scales linearly in the number of observations. Building on these ideas, we propose two conditional NPs (CNPs): SFConvCNPs, which stack SFConv blocks with non-linearities, and SFVConvCNPs, which integrate the Volterra formulation. Experiments on synthetic and real-world datasets demonstrate our methods' efficacy against state-of-the-art baselines.

Peiman Mohseni, Nick Duffield, Raymond K. W. Wong

Texas A&M University, College Station, Texas, USA.

Neural ProcessesFrequency-Domain Feature LearningEquivariant Neural NetworksNeural Operators
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