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.

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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.

Aug 3, 2026

tFUSOperator: Operator Learning for Transcranial Focused Ultrasound Digital Twins

Transcranial focused ultrasound (tFUS) requires accurate estimation of the intracranial acoustic field, which is distorted by skull-induced aberrations. Numerical solvers are accurate but computationally expensive for digital twins, where the field must be re-estimated repeatedly as treatment conditions change. Existing deep-learning surrogates are fast but typically use voxel-to-voxel regression on a fixed grid, with no mechanism reflecting how acoustic energy propagates through the skull. We instead cast tFUS simulation as an operator learning problem and propose tFUSOperator, a coordinate-aware neural operator that maps the free-field pressure, skull anatomy, and treatment parameters to the intracranial field within a shared physical coordinate frame. To our knowledge, this is the first operator-based formulation of tFUS field prediction. On both seen and unseen skulls, the model localizes the acoustic focus accurately-reaching about 90% and 72% Dice, respectively-and it performs nearly as well from magnetic resonance (MR) as from computed tomography (CT) input while running 5.6×1045.6 \times 10^4 times faster than numerical simulation. These results suggest a fast, radiation-free route to safe and practical digital twins for patient-specific tFUS treatment. The code is available at: https://github.com/CMME-Lab/tFUSOperator.git.

Minjee Seo, Haris Ghafoor, Minju Seol +2

School of Mathematics and Computing (Computational Science and Engineering), Yonsei University, Seoul, Republic of Korea · Innovative & Intelligent Computational Science Institute (IN2CSI), Seoul, Republic of Korea

PDE Operator LearningDigital TwinsNeural Operators
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Aug 2, 2026

Amortizing the Calibration Triple: A Projection-Consistent Neural Operator for Local-Stochastic Volatility

Local-stochastic volatility (LSV) combines vanilla marginals with richer smile dynamics, but calibration requires a slow, noisy and sequential McKean--Vlasov fixed point. We learn a projection-consistent operator for the calibration triple. Given finite quotes and a stochastic-volatility (SV) backbone, it jointly returns an implied-volatility surface subject to static-arbitrage constraints, its Dupire local volatility, LSV leverage and the conditional moment required by the projection identity. Starting from option-price marginals, we derive a division-free Dupire residual in log-implied-variance coordinates and a quotient Fokker--Planck equation after Gyöngy projection. Deep Operator Network (DeepONet) and Fourier Neural Operator (FNO) implementations enforce quote fit, static-arbitrage, Dupire and projection constraints. For the witness-augmented residual system, we prove conditional identification and empirical consistency under LSV existence and inverse residual stability. In controlled synthetic tests, forward-start and cliquet errors differ from a particle method by 0.1 and 0.2 percentage points, while calibration latency falls from 98.5 to 0.6 ms. Compared with the tested baselines, local-volatility root-mean-square error (RMSE) falls by 36% and leverage RMSE by 7-16%. These results support amortizing the LSV fixed point: the expensive solve moves offline, while online calibration reduces to a single projection-consistent operator evaluation.

Xiaozhen Wang, Anaïs Després, Martin Dureau +1

1CEREMADE, Université Paris Dauphine-PSL · 2LaMME, Université Évry Paris-Saclay · 3AIML Global Markets, Barclays

Quantitative FinanceModel CalibrationNeural OperatorsFinance
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Aug 1, 2026

CoSynFlow: Conformal Symplectic Neural Flows for Cross-System Prediction of Dissipative Hamiltonian Dynamics

Learning solution operators for differential equations is a central problem in scientific machine learning. However, many neural operator methods optimize prediction accuracy without explicitly enforcing the geometric structure of the dynamics. Structure-preserving models such as SympNets and Symplectic Neural Flows address this issue for conservative Hamiltonian systems by preserving the symplectic form. In dissipative Hamiltonian systems with conformal symplectic structure, however, the symplectic form evolves according to a conformal factor determined by the dissipation. We propose CoSynFlow, a conformal symplectic neural flow for learning continuous-time solution maps of dissipative Hamiltonian dynamics. CoSynFlow composes symplectic shear maps with explicit conformal scaling, preserving the conformal symplectic structure by construction. By conditioning it on a finite-dimensional Hamiltonian descriptor and the dissipation parameter, a single trained model predicts solution maps for unseen systems without retraining. CoSynFlow keeps the structure error at machine precision, attains the lowest long-horizon error, and admits physics-informed training.

Baige Xu, Takaharu Yaguchi

RIKEN AIP · Motooka Nishi-ku, Fukuoka, Japan · IMI, Kyushu University

Neural Operators
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Jul 31, 2026

Neural Operator Learning for Collision-Aware Trajectory Planning of Spacecraft Swarms

Satellite constellations require orbital transfers that are both fuel efficient and collision avoidant. Yet, the computational cost of optimization methods traditionally used to plan their trajectories scales poorly with both the number of satellites as well as the number of obstacles to avoid, due to the pairwise safety constraints. In this work, we introduce a permutation-equivariant neural operator for trajectory planning of spacecraft swarms. This neural operator maps distributions of spacecraft initial states, target states, and obstacle initial states to trajectories which avoid collision and conserve fuel. This neural operator output is then paired with a batched Gauss-Newton finish to enforce exact orbital dynamics, and further reduce fuel use. The operator is self-supervised, trained without optimal trajectory labels. When trained on ten spacecraft, the proposed method generalized zero-shot to swarms of 1,000 spacecraft and 11,000 obstacles. The generated trajectories matched a per-agent optimal control solver's accuracy while retaining collision avoidance. Operator learning grounded in physics may offer a fast, scalable alternative to trajectory optimization in the increasingly crowded orbits of the future.

Sidhdharth D. Sikka, Suyi Gao, Zehui Lu +2

Manifold Research Group and the School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907 USA · Department of Mathematics, Purdue University, West Lafayette, IN 47907 USA · Independent Researcher, Fremont, CA USA

Collision AvoidanceTrajectory OptimizationNeural Operators
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Jul 31, 2026

HERO: History-Enriched Rollout Training for Long-Horizon Autoregressive Neural Operators

Neural operators provide fast surrogates for time-dependent partial differential equations (PDEs) by applying a learned evolution operator recursively to its own predictions, but this autoregressive rollout feeds every prediction error back as input, so local errors accumulate. Existing rollout-training strategies reduce the mismatch between training inputs and self-generated states, yet their supervision still measures only the absolute discrepancy from the ground-truth trajectory. Such supervision is therefore uninformative about whether the operator has overcome the long-horizon failure behaviors it exhibited earlier during optimization. We propose history-enriched rollout training (HERO), which augments conventional absolute trajectory supervision with relative supervision derived from the model's optimization history. HERO ranks detached candidate rollouts from a periodically refreshed lagged operator, the current model, and a perturbed input by rollout error, spectral discrepancy, energy drift, and error growth, and selects the strongest failure trajectory as reference. This reference enters a margin-based objective as a fixed comparison baseline, inducing a bounded, sample-dependent reweighting of the ground-truth rollout gradient rather than an independent gradient direction, which we further analyze theoretically. Experiments on nine PDE benchmarks with spectral and attention-based backbones show that HERO consistently improves long-horizon accuracy, stable rollout length, and out-of-distribution robustness at no inference-time cost. These results indicate that history-enriched relative supervision is effective for stabilizing long-horizon autoregressive prediction.

Jiaquan Zhang, Shuxu Chen, Haifan Meng +6

School of Computer Science and Engineering, UESTC · Electronics and Information Convergence Engineering, KHU · Department of Mathematical Sciences, UOL +3

Neural Surrogate ModelingPDE Operator LearningNeural Operators
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Jul 31, 2026

Geometry-aware Incremental Neural Operator for Long-Horizon PDE prediction

Neural operators have shown strong potential for learning solution operators of partial differential equations (PDEs). However, long-horizon autoregressive prediction remains challenging: local errors accumulate as spectral inconsistency, phase misalignment, or mean drift. Existing methods mainly improve state representations and operator backbones, while leaving the repeatedly applied latent transition increment weakly structured, allowing spectral errors and unstable channel couplings to accumulate during rollout. To address these issues, we propose a geometry-aware incremental neural operator (GeoIncNO) for stable long-horizon PDE prediction. GeoIncNO predicts latent increments for residual advancement and uses lightweight low-rank projectors to regulate channel coupling within active frequency bands derived from the increment spectral energy distribution. To reduce physical-space reconstruction errors, GeoIncNO further introduces a mean--fluctuation decoupled reconstruction mechanism, where stable mean structures and dynamic fluctuations are fused separately, and phase correction is applied only to the zero-mean fluctuation component. Extensive experiments on six PDE benchmarks, covering 1D, 2D, and 3D dynamical systems, show that GeoIncNO achieves consistently strong prediction accuracy, improved rollout stability, and better spectral fidelity compared with competitive neural-operator baselines.

Jiaquan Zhang, Shuxu Chen, Haifan Meng +6

School of Information and Software Engineering, UESTC · Electronics and Information Convergence Engineering, KHU · School of Computer Science and Engineering, UESTC +4

PDE Surrogate ModelingPDE Operator LearningSpatiotemporal ForecastingNeural Operators
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Jul 30, 2026

Orca: Neural Operators for Causal Reasoning in Continuous Time

Structural causal models are the standard language for reasoning about interventions and counterfactuals, but they describe static variables, typically measured once, and usually forbid cyclic dependencies. Many systems we care about, such as patients, climates, and economies, instead evolve continuously in time, are observed at irregular time points, and contain feedback loops. We argue that neural operator learning provides a natural foundation for causal reasoning in this setting, and propose Orca, a framework in which each node of the causal graph is a function of time and each mechanism is a learned map between function spaces. We extend existing neural operator architectures to express causal mechanisms: a mechanism computes the function value of a node from its parent nodes by taking several parent functions as input, respects the arrow of time, and treats latent exogenous noise as a function that can be inferred and reused for counterfactuals. We formalize the model class and demonstrate counterfactual reasoning on synthetic continuous-time examples. Code is available at https://github.com/gerritgr/orca

Gerrit Großmann, David A. Selby, Sebastian J. Vollmer

German Research Center for Artificial Intelligence (DFKI), Kaiserslautern, Germany

Structural Causal ModelsCausal Counterfactual GenerationCausal InferenceCounterfactual ReasoningNeural Operators
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Jul 29, 2026

Inverse Learning of Latent Risk-Neutral Densities from Irregular Option Quotes

Accurate option prices do not imply accurate recovery of the latent risk-neutral density. We study this distinction with two complementary benchmarks. A controlled benchmark exposes simulator-truth densities for latent evaluation, while a chronological NIFTY benchmark tests only held-out market prices. A two-component lognormal mixture has the lowest aggregate price, L1L^1, Wasserstein, and fixed-tail errors on the synthetic benchmark. Learned operators retain narrower strengths: DeepONet reduces 1% quantile and variance error by 39.0% and 34.6% relative to the mixture, and a quote transformer reduces L1L^1 by 16.4% on the structurally misspecified Merton family. A numerical conditioning analysis explains why these rankings can differ: after enforcing mass and forward constraints, 95 of 126 pricing directions are numerically null, and two densities separated by L1=0.061L^1 = 0.061 produce identical prices on the covered strikes. On 524 held-out NIFTY calls, validation-selected test-time adaptation reduces DeepONet RMSE by 28.3%, but per-expiry mixture and SVI fits remain much more accurate. The evidence supports target-dependent inductive bias, not a universal winner.

Lennon J. Shikhman, Michael Galarnyk, Aadi Dash +1

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

Quantitative FinanceDeep Operator NetworksDensity EstimationNeural OperatorsInverse ProblemsFinance
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Jul 26, 2026

No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study

We test whether an architecture that succeeds on a simple flow regime also succeeds on a richer one, with each trained separately on each regime. We explore two transient flows under time-varying boundary conditions: the three-dimensional slurry film in chemical-mechanical planarisation (CMP), central to semiconductor manufacturing, and the two-dimensional Kármán vortex street (KVS). Eight surrogate models on one shared pipeline differ in whether they learn the full field or a latent representation, and in whether they predict in one shot or step by step. No single architecture wins both regimes. On the film, a one-shot full-field model reconstructs the cumulative wall shear stress to 2.7% relative error. On the wake, a latent autoregressive DeepONet retains 90% of the shedding power that direct and one-shot models damp to almost zero. The treatment of time decides the outcome. The self-sustained wake calls for autoregressive feedback and the boundary-driven film for a direct map. Pointwise RMSE hides the damped oscillation on the wake, compresses the sixfold lead on the film's process target, and picks the damped model under wake extrapolation. The evaluation scores five physical questions. Trained surrogates answer queries 10^3 to 10^4 times faster than the finite-element solver and pay off from the first query beyond the training set on the film and from the third on the wake. Neither the winning architecture nor its validation holds across regimes. The choice of surrogate should follow the dynamical character of the target flow, and its validation should resolve the failure modes.

Georg Winkler, Martin Stoll

Department of Mathematics, Chemnitz University of Technology, 09111, Chemnitz, Germany

Neural Surrogate ModelingPDE Surrogate ModelingNeural OperatorsComputational Fluid DynamicsAutoregressive Modeling
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Jul 25, 2026

Neural operator discovery from heterogeneous trajectories

Neural operators provide data-driven mappings for modeling dynamical systems. Extending them to families of systems typically requires explicit conditioning variables such as physical parameters, geometries, or boundary conditions. In many real-world settings, these quantities are unobserved. Here, we formulate neural operator discovery (NOD) as the problem of learning both shared solution operators and system-specific variation directly from heterogeneous trajectories without access to labeled governing factors. We introduce a factorized latent-conditioning formulation that jointly learns a neural operator and a low-dimensional latent representation through factorized prediction, trajectory-decoupled sampling, and dimension selection. Across diverse systems, the learned latent representation captures the intrinsic dimensionality of system variation and organizes system instances in a smooth and approximately invertible latent structure aligned with the underlying governing factors. This organization enables generalization to previously unseen system instances, including zero-shot extrapolation across regimes and stable long-horizon prediction. These results establish an interpretable paradigm for operator learning in the absence of explicit factor supervision.

Zituo Chen, Qiaofeng Li, Jiaxin Hu +1

Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, 02139, MA, USA. · State Key Laboratory of Fluid Power & Mechatronic Systems, Zhejiang University, 866 Yuhangtang Road, Hangzhou, 310058, Zhejiang Province, China.

Latent Dynamics ModelingNonlinear System IdentificationOperator LearningNeural Operators
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Jul 24, 2026

Generalized Neural Operator for Parametric and Boundary-Value Problems

Developing foundational neural simulators for Partial Differential Equations (PDEs) requires robust generalization across diverse physical parameters and boundary conditions. However, current deep learning approaches largely face a structural trade-off between condition-agnostic deployment and physical fidelity. Purely data-driven operators infer the underlying physics implicitly and thus lack the explicit constraints needed to ensure physically valid solutions across varying domains, rendering the learning problem ill-posed. On the other hand, Physics-Informed Neural Networks (PINNs) enforce rigorous physical constraints but necessitate costly, instance-specific optimization. Furthermore, the massive scale of emerging foundational operators has severely degraded their inference speeds, making them computationally uncompetitive with traditional numerical solvers. To address this bottleneck between condition-agnostic deployment, physical rigor, and inference efficiency, we propose a Generalized Neural Operator. By formalizing the classical conditions for well-posedness within neural operators, our framework demonstrates the theoretical benefits of explicitly conditioning on PDE parameters and boundary conditions. To implement this synthesis without compromising computational speed, we introduce three novel architectural components: a parameter-gated mixture of kernels for efficient parameter generalization, a generalized boundary transfer operator that projects arbitrary boundary constraints into a unified latent Dirichlet representation, and a specialized training objective to ensure stability. Extensive experiments demonstrate that our theoretically grounded approach achieves superior generalization across heterogeneous physical regimes while maintaining strict inference efficiency comparable to conventional numerical baselines.

Ruoyan Li, Yizhou Sun, Wei Wang

University of California, Los Angeles

Efficient Neural Network InferencePDE SolvingPDE Operator LearningPhysics-Informed Neural OperatorsNeural Operators
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Jul 21, 2026

Countercurrent Multiplier Networks: A Renal-Inspired Iterative Operator with Provably Bounded Fixed-Point Dynamics

The mammalian kidney concentrates urine using a mechanism with no analogue in current neural architectures: the countercurrent multiplier. Two anti-parallel flows joined at a hairpin recirculate a weak magnitude-bounded local pump into a large axial gradient achieving a four-fold concentration increase from a single-effect gradient that never exceeds 200 mOsm at any point. We formalize this mechanism as a differentiable sequence operator the Countercurrent Multiplier (CCM) layer and study it as an alternative to residual iterative refinement.

Snigdha Chandan Khilar

Independent Researcher

Dynamical SystemsIterative RefinementNeural Operators
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Jul 20, 2026

Adaptive Mamba Neural Operators

Accurately solving partial differential equations (PDEs) on arbitrary geometries and a variety of meshes is an important task in science and engineering applications. In this paper, we propose Adaptive Mamba Neural Operators (AMO), which integrates reproducing kernels for state-space models (SSMs) rather than the kernel integral formulation of SSMs. This is achieved by constructing Takenaka-Malmquist systems for the PDEs. AMO offers new representations that align well with the adaptive Fourier decomposition (AFD) theory and can approximate the solution manifold of PDEs on a wide range of geometries and meshes. In several challenging benchmark PDE problems in the fields of fluid physics, solid physics, and finance on point clouds, structured meshes, regular grids, and irregular domains, AMO consistently outperforms state-of-the-art solvers in terms of relative L2L^2 error. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.

Zeyuan Song, Zheyu Jiang

School of Chemical Engineering Oklahoma State University Stillwater, OK 74078, USA

PDE SolvingPDE Operator LearningNeural OperatorsState Space ModelsPartial Differential Equations
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Jul 15, 2026

NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes

We address fundamental challenges in representing and computing Rd\mathbb{R}^{d}-valued predictable square-integrable processes over [0,T][0,T], collected in the space HT2(Rd)\mathcal{H}^2_T(\mathbb{R}^{d}). These processes are central to continuous-time stochastic control, reinforcement learning, and mathematical finance. Although Wiener-chaos expansions offer strong theoretical tools, traditional computational methods are hindered by the need for large chaos dictionaries and high-order iterated integrals. To overcome these obstacles, we introduce NeuralChaos -- a neural operator architecture that produces elements of HT2(Rd)\mathcal{H}^2_T(\mathbb{R}^{d}) using only finitely many evaluations of the driving Brownian motion, while preserving predictability and square-integrability. We prove that NeuralChaos is dense in HT2(Rd)\mathcal{H}^2_T(\mathbb{R}^{d}) and achieves the best NN-term chaoslet approximation rates for compressible and Malliavin--Sobolev regular processes. Moreover, compressibility is shown to be typical for processes from HT2(Rd)\mathcal{H}^2_T(\mathbb{R}^{d}) under non-degenerate sub-Gaussian sampling. In contrast, we show that finite-dimensional Markovian neural SDE models constitute a meagre and Gaussian-null subset in HT2(Rd)\mathcal{H}^2_T(\mathbb{R}^{d}), regardless of discretization, whereas compressible processes are generic. Numerical experiments on a stochastic optimal control problem and dynamic hedging highlight the practical effectiveness of our approach. Our results enable more efficient and expressive modelling in stochastic analysis and mathematical finance.

Anastasis Kratsios, Giulia Livieri, Philipp Schmocker

Neural Network Approximation TheoryNeural Operators
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Jul 14, 2026

Deep Learning-based Surrogate Modelling of the LOD Method for Multiscale Problems

Multiscale problems are notoriously difficult to tackle using traditional numerical methods, as accurately resolving fine-scale features often requires prohibitively fine discretizations. This challenge is particularly pronounced in applications such as materials science, fluid dynamics, climate systems, chemical processes, and complex networks. Recent neural operator models provide a promising data-driven alternative, but frequently struggle to achieve sufficient accuracy in the presence of strongly heterogeneous or oscillatory coefficients. In this work, we focus on the solution of elliptic PDEs with rough and high-contrast inputs. The Localized Orthogonal Decomposition (LOD) method is a well-established numerical approach for such problems, but it comes, however, at a substantial computational cost. We investigate the performance of popular neural operator architectures on these challenging multiscale problems and identify key limitations in their ability to resolve fine-scale structure. To overcome these challenges, we introduce LOD-MSNO (LOD-Multiscale Neural Operator), a hybrid approach that leverages the LOD method as a strong multiscale prior by building on its representation of the solution as a linear combination of problem-adapted basis functions, while addressing its main computational bottlenecks through data-driven operator learning. We further provide theoretical error estimates for the proposed coefficient-learning framework. Lastly, we demonstrate the potential of our proposed method to outperform current neural operator baselines in terms of accuracy for challenging multiscale inputs, while mainly retaining the computational efficiency of neural operator models.

Marc Haltmayer, Jaemin Seo, Yuseung Lee +3

Department of Artificial Intelligence, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul, 06974, Republic of Korea · Computational Science and Engineering Team, Samsung Electronics, 41 Samsung-ro, Giheung-gu, Yongin-si, 17113, Republic of Korea

PDE SolvingPDE Surrogate ModelingPDE Operator LearningNeural Operators
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Jul 13, 2026

An Agentic AI Scientific Community for Automated Neural Operator Discovery

We present an agentic approach to autonomous neural operator discovery based on an AI scientific community, which consists of a swarm of virtual laboratories that interact under a citation-based economy of influence. Highly-cited labs found new labs that follow their research direction and replace non-performing labs. Each virtual lab contains three agents: an LLM planner that proposes an architecture, a numerical worker that trains and measures it, and an LLM reviewer that participates in cross-lab peer review. All labs share a common vocabulary consisting of DeepONet (branch-trunk), Fourier, Transformer (attention), wavelet, and residual convolutional neural operator building blocks. We evaluate the neural operator AI scientific community on five problems, namely piecewise regression, the linear advection and Burgers 1D PDEs, and the Navier-Stokes and Darcy flow 2D PDEs, while repeating the simulation three times for each problem. The results show that the neural operator AI scientific community is capable of discovering high-accuracy, low-parameter-count neural operator architectures. All 9,623 LLM calls are logged and audited, which reveals that the virtual lab LLM planners choose to hybridize in 99.8% of their logged decisions, consistently returning multi-family hybrids. Moreover, we conducted an ablation study by replacing the LLM agents in each lab by rule-based alternatives, which caused the scientific community to collapse to non-hybridized single-family stacks in several cases, showing that LLM agency is needed to preserve diversity. The results suggest a no-free-lunch theorem for neural operators: there is no universal winner. The code, configurations, and the complete LLM transcripts are released at https://github.com/luislootx/AI-SC.

Luis Loo, Ulisses Braga-Neto

Department of Electrical & Computer Engineering Texas A&M University, College Station, TX

AI Agents for Scientific DiscoveryLLM Agent OrchestrationAutonomous Scientific DiscoveryPDE Operator LearningNeural Architecture SearchNeural Operators
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Jul 9, 2026

PGD-NO: A Neural Operator with Precomputed Geometry Decomposition for 3D Million-scale Physics Simulations

While neural PDE solvers have demonstrated significant potential for accelerating engineering simulations, existing architectures remain constrained by high memory consumption and the single node bottleneck, where the maximum processable mesh resolution is strictly limited by the VRAM of a single compute unit. To address these challenges, we propose PGD-NO, a neural operator with Precomputed Geometry Decomposition, that relocates the computational overhead of geometric encoding to a deterministic pre-computation phase. By utilizing an iterative geometry decomposition algorithm to extract geometry tokens, our model decouples feature extraction from solution querying. This architecture enables linear memory scalability, allowing high fidelity learning on meshes exceeding 10 million nodes, a scale where existing architectures typically encounter memory exhaustion. PGD-NO demonstrates competitive predictive accuracy across diverse industrial benchmarks and provides intrinsic interpretability through attention mechanisms. By effectively overcoming traditional mesh-size constraints, PGD-NO offers a robust and efficient solution for the next generation of large-scale, high-fidelity industrial design applications.

Weiheng Zhong, Jing Bi, Victor Oancea +1

Department of Civil and Environmental Engineering, Grainger College of Engineering, University of Illinois Urbana-Champaign, Urbana, IL, US · Dassault Systemes Americas, Johnston, RI, US · Siebel School of Computing and Data Science, Grainger College of Engineering, University of Illinois Urbana-Champaign, Urbana, IL, US

PDE Surrogate ModelingPDE Operator LearningNeural Operators
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Jul 8, 2026

Hybrid Least Squares/Gradient Descent Methods for MIONets

In this paper, we propose an efficient hybrid least squares/gradient descent (LSGD) method for MIONets to accelerate training. This method generalizes the LSGD method for DeepONets. Since MIONet is the sum of the entrywise product of multiple branch networks and a trunk network, it can be viewed as a multilinear function with respect to the last layer parameters of each branch network. These sets of parameters can be optimized using the alternating least squares method, where we solve the LS system for a single branch network in turn. To handle the large-sized system matrix, we introduce Kronecker and Khatri-Rao products and tensor permutation matrices to factor the large matrix into small ones. Our method is compatible with a general type of L2L^2 loss with regularization terms for the last layer parameters of each branch, where linear operators can be applied to the MIONet output in each loss term.

Jun Choi, Chang-Ock Lee, Minam Moon

Department of Mathematical Sciences, KAIST, Daejeon 34141, KOREA · Department of Mathematics, Korea Military Academy, Seoul 01805, KOREA

Gradient DescentNeural Network OptimizationDeep Operator NetworksOperator LearningNeural Operators
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Jul 7, 2026

Knowledge-Constrained Shape Optimization with a Mixture-of-Experts Neural Operator for High-Confidence Design

Engineering shape optimization faces challenges in both expert-dependent problem setup and surrogate-model reliability. In practical aerodynamic design, optimization settings such as editable regions, deformation ranges, and design-preservation constraints are typically specified manually by experienced engineers, while surrogate-based optimization may become unreliable for heterogeneous geometry databases and out-of-distribution designs. To address these challenges, we propose a knowledge-constrained shape-optimization framework that translates knowledge-based constraints and user intent into quantifiable parameters of DFFD-based deformation operators, enabling engineering-aware and controllable constrained optimization. We further develop a Mixture-of-Experts Neural Operator (MoE-NO) to improve drag prediction and trend consistency over heterogeneous aerodynamic datasets. Based on the MoE-NO encoder and Mahalanobis distance, an uncertainty-estimation strategy is introduced to detect out-of-distribution geometries and selectively trigger physics-solver feedback for local sample enrichment. Experiments on in-house MPV, SUV, and Sedan datasets show that MoE-NO achieves a test-set MAPE of 1.16%1.16\% and a trend-prediction accuracy of 94.34%94.34\%, outperforming the best baseline results of 1.52%1.52\% and 90.34%90.34\%, respectively. Vehicle shape-optimization experiments further yield CFD-validated drag coefficient reductions of approximately 4%4\% to 10%10\%.

Wenhao Fan, Yuanwei Bin, Jianghan Gu +4

School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China · 2Eastern Institute of Technology, Ningbo, Ningbo 315200, Zhejiang, China · 3TenFong Technology Co., Ltd., Shenzhen 518000, Guangdong, China +1

Neural Surrogate ModelingAutomotive AerodynamicsConstrained OptimizationNeural Operators
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Jul 5, 2026

Quadrature-Aware Complex-Linear Neural Operator for Boundary-to-Field Prediction in Resonant Acoustics

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

Muhammad Idrees Khan, Hua-Dong Yao

Department of Mechanical Engineering, Chalmers University of Technology Gothenburg 41296, Sweden

Complex-Valued Neural NetworksOperator LearningPhysics-Based SimulationNeural Operators
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Jul 4, 2026

LLT: Local Linear Transformer for PDE Operator Learning

Neural operators have become a common approach for learning PDE solution maps and accelerating numerical simulations. Transformer-based neural operators are of particular interest, since attention can learn long-range dependencies in the computational domain. However, standard attention has two major limitations when applied to PDEs: it scales quadratically with the number of computational nodes, and it lacks an explicit bias toward local interactions. To address these issues, we introduce Local Linear Transformer (LLT) for PDE operator learning. The architecture combines linear global attention with local spatial mixing, and incorporates coordinate and geometry information. We evaluate LLT on several PDE problems, including elasticity, plasticity, airfoil flow, pipe flow, and Darcy flow. The reference data for these problems span finite-element, finite-volume, and finite-difference discretizations on structured and unstructured meshes. Compared with other neural-operator and transformer baselines from prior studies, LLT achieves competitive or lower relative L2L_2 error across these problems. On matched structured discretizations, wall-clock time per training iteration is reduced by factors of 1.8 to 2.5 relative to Transolver. We also scale the approach and apply it to a three-dimensional car aerodynamics dataset with 32,186 unstructured mesh points per sample. Together, these results indicate that LLT provides an accurate and computationally efficient operator for PDE problems across discretizations, mesh types, and problem settings.

Oded Ovadia, Eli Turkel

School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

PDE Surrogate ModelingTransformer AttentionLinear AttentionPDE Operator LearningNeural OperatorsTransformer Neural Operator
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Jul 2, 2026

LiNO: Lifting based multiresolution neural operator

Recently, neural operators have shown promising outcomes for learning solution operators of differential equations directly from data. This framework learns a functional mapping from the parameter field to the solution field, enabling the prediction of an entire class of solutions rather than a specific instance. However, existing operators often struggle to capture both global dynamics and fine-scale structure simultaneously. To design an effective operator capable of representing multiscale features, a hierarchical multiscale decomposition framework is required. In this study, we develop the Lifting Neural Operator (LiNO), a multiresolution operator built on the second-generation wavelet lifting scheme. LiNO learns a multiresolution decomposition directly from data by parameterizing the lifting transform. This lifting transformation is adaptive to the underlying solution function and exactly invertible by construction, enabling information-preserving multiscale operator learning. In the lifted multiresolution space, the operator evolves coarse and directional detail coefficients separately, resulting in scale-aware modeling of the underlying physics. We evaluate LiNO on several benchmarks, including Darcy flow, the Poisson equation, the Allen-Cahn equation, the compressible Navier-Stokes equation, and the Gray-Scott reaction-diffusion system. Together, these benchmarks cover a wide range of physical behaviors, including multiscale phenomena, transport-dominated dynamics, and chaotic systems. LiNO demonstrates strong performance on these challenging benchmarks compared with state-of-the-art neural operators. These results suggest that adaptive multiresolution operators provide a promising direction for scientific machine learning.

Himanshu Pandey, Subham Patel, Ratikanta Behera

Department of Computational and Data Sciences, Indian Institute of Science, Bangalore 560012, India · IISc Mathematics Initiative, Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

Wavelet MethodsPDE Operator LearningNeural Operators
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Jul 2, 2026

Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data

Operator learning has emerged as a powerful tool for modeling complex physical systems in functional spaces. However, their neural network-based architectures make them opaque models, obscuring the reasoning behind their predictions. In this work, we introduce a self-explainable operator learning framework that overcomes this challenge by reformulating operator learning as a linear combination of generalized functional linear models expressed through integral equations. Exploiting the additive decomposability of these integral equations, we divide the input domain into subdomains and compute localized integrals to evaluate the contribution of each region to the final prediction. This decomposition enables direct interpretability where the model explains both inputs and outputs by linking specific input regions to corresponding output patterns, thereby revealing which spatial features drive predictions. We demonstrate the framework on function-to-scalar and function-to-function mappings in fluid flow problems involving blood flow and unsteady aerodynamics. The results show that the operator most often prioritizes regions with strong feature gradients, providing physically meaningful insight into the model's decision-making process. Comparisons with established post-hoc explainability methods demonstrate qualitative agreement while highlighting the key advantage of the proposed approach: explainability is embedded directly within the operator structure itself and does not require an external tool. Therefore, our framework provides a mathematically transparent and physically interpretable approach to uncover relationships within data, fostering trust in machine learning for scientific applications by enabling more informed data-driven analysis of physical systems.

Mojgan Alishiri, Amirhossein Arzani

1Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT, 84112, United States · 2Mechanical Engineering Department, University of Utah, Salt Lake City, UT, 84112, United States

Neural Network InterpretabilityOperator LearningNeural Operators
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Jul 1, 2026

GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems

Operator learning for partial differential equations (PDEs) on arbitrary geometries builds fast neural surrogates for large-scale simulation. Although recent geometry-adaptive neural operators have made substantial progress, they are mainly designed for forward problems in which inputs and outputs share the same spatial domain. This limits their applicability for boundary value problems (BVPs) and inverse problems, where inputs and outputs may live on different domains. We introduce the Geometry-Adaptive Integral Autoencoder (GAIA), an operator learning model that encodes the domain boundary and the interior field distribution into geometry tokens, and conditions integral transform layers on these tokens via cross-attention, allowing the kernel to adapt locally to geometric features. This yields a single architecture for forward (including BVPs) and inverse problems on arbitrary domains in one pass, without retraining, iterative optimization, or graph construction. We evaluate GAIA on seven 2D and 3D benchmarks, four of which are new or substantially extended benchmarks for inverse problems and BVP: electrical impedance tomography, optical tomography, 3D Darcy flow on varying geometries, and a modified setting of Poisson BVP on mechanical components benchmark (MCB). GAIA sets new state-of-the-art results on every inverse and BVP task, reducing median relative L2L^2 error by 64% on airfoil flow reconstruction and 27% on EIT relative to the next best amortized method, and outperforming all baselines on every shape category of MCB. On other forward problems, GAIA is competitive with specialized solvers while maintaining stable accuracy across point resolutions on which transformer-based baselines degrade.

Meenakshi Krishnan, Pranav Pulijala, Ke Chen +2

University of Maryland, College Park · University of Delaware, Newark

PDE SolvingImage ReconstructionPDE Operator LearningPDE Inverse ProblemsNeural Operators
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Jul 1, 2026

From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators

We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-TT solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier--Stokes, Allen--Cahn, and Cahn--Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.

Nisha Chandramoorthy, Daniel Sanz-Alonso, Nathan Waniorek

University of Chicago

Dynamical SystemsFourier Neural OperatorPDE Operator LearningNeural Network Approximation TheoryNeural Operators
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Jun 28, 2026

Randomized neural operator for parametric PDEs with fast training and conformal uncertainty quantification

Repeatedly solving parametric PDEs is essential for uncertainty quantification, design optimization and inverse problems, but conventional neural operators require expensive non-convex training. We introduce PCA--RaNN, a randomized latent neural operator that combines PCA-based dimensionality reduction with fixed random features and a closed-form least-squares readout. It recasts latent operator learning as fixed-feature linear regression, reducing training time by one to three orders of magnitude across benchmarks while maintaining competitive accuracy. We introduce an energy-matched scaling rule and a lightweight two-parameter BFGS refinement to correct suboptimal feature scales. Ensemble averaging reduces predictive variance. On Burgers, Darcy, Navier--Stokes and backward heat equation benchmarks, PCA--RaNN provides a favorable speed--accuracy trade-off against operator-learning baselines. The ensemble supports split-conformal prediction intervals, and the linear readout enables rapid online adaptation via recursive least squares without retraining hidden features. This provides an efficient, uncertainty-aware surrogate for many-query scientific workflows.

Zirui Deng, Jingbo Sun, Deyu Meng +1

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

Uncertainty QuantificationPDE Surrogate ModelingPDE Operator LearningNeural Operators
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Jun 26, 2026

OperatorSHAP: Fast and Accurate Shapley Value Estimation for Neural Operators

Understanding model predictions is essential for physical applications, where outputs often inform safety-critical decisions, such as structural load assessment, weather warnings, and clinical diagnosis. Shapley values satisfy many desirable properties as an attribution method, but their computational cost during inference hinders their practical use. Current amortized explainers, such as FastSHAP, are limited to homogeneous inputs, which is problematic for physical applications where data often comes from irregular grids and geometries. We introduce OperatorSHAP, a grid-agnostic attribution method and training procedure that allows us to train FastSHAP-like explainers for neural operators. We establish a theoretical framework for attributions in function space, connecting to Aumann-Shapley values. We further show that OperatorSHAP's explanations are consistent with state-of-the-art discrete Shapley values across resolutions and transfer across grid sizes without retraining.

Joshua Stiller, Santo M. A. R. Thies, Felix Czaja +1

LMU Munich, MCML · LMU Munich · LMU Munich, MCML, DFKI

Shapley Value AttributionNeural Operators
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Jun 23, 2026

Real vs. Complex Spectral Bases for Neural Operators: The Role of Green's Function Alignment

Fourier Neural Operators (FNO) learn solution operators of partial differential equations by parameterizing global convolutions in the complex Fourier domain. For real-valued PDE solutions, the complex FFT carries representational redundancy through conjugate symmetry. We introduce the Hartley Neural Operator (HNO), the exact real-valued mirror of FNO: it replaces the FFT with the purely real Discrete Hartley Transform and learns a single real multiplier per retained spectral mode, with no complex arithmetic. Because the real Hartley spectrum is not halved by conjugate symmetry, HNO retains twice as many frequency corners as FNO but one real weight where FNO carries a complex pair, so the two operators are iso-parametric at equal width and differ only in spectral basis. Our central thesis is that the best basis is a property of the operator. Self-adjoint elliptic operators (Poisson, biharmonic) have real, symmetric Green's functions that the real Hartley multiplier diagonalizes exactly, and HNO is favored there. Time-dependent operators carry phase, from oscillation in the wave equation to transport in advection, Burgers, and Navier-Stokes, which a real diagonal multiplier cannot represent, so FNO is favored there, and increasingly so with the operator's phase content, leaving the phaseless heat equation as the borderline case. Training both operators identically and benchmarking across PDE classes, initial-condition families, and boundary conditions, we find an elliptic-versus-time-dependent split that is monotone in operator phase content and matches the Green's-function theory we develop. Rather than a universal winner, our findings give a predictive rule: match the spectral basis to the symmetry of the solution operator.

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 OperatorPDE Operator LearningSpectral MethodsNeural Operators
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Jun 22, 2026

The Fractal Neural Operator: Overcoming Spectral Bias in Chaotic Attractors via Prime-Harmonic Weierstrass Encodings

Deep learning models, particularly Transformers and Neural Operators, exhibit a well-documented "spectral bias," effectively acting as low-pass filters that smooth out high-frequency information. While benign in fluid dynamics, this bias is catastrophic for Chaotic Dynamical Systems, where the underlying strange attractor is characterized by fractal geometry and infinite spectral density. We introduce the Fractal Neural Operator (FNO), a novel architecture that utilizes a non-resonant prime number basis to approximate continuous dynamical systems. Unlike geometric encodings (2k2^k), which suffer from spectral gaps and resonance, our Harmonic Weierstrass Encoder injects infinite spectral resolution into the latent space. We demonstrate that FNO extends the valid prediction horizon of the Lorenz-63 system to 347 Lyapunov times, exceeding state-of-the-art Reservoir Computing baselines by a factor of 2.3x. These results suggest that "chaos" is not inherently unpredictable to neural networks, but rather requires non-differentiable, fractal embedding manifolds.

Kanishk Awadhiya

Independent Researcher

Spectral BiasChaotic Dynamical SystemsNeural OperatorsChaotic Time-Series Prediction
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Jun 22, 2026

Neural Operator Processes for Probabilistic Operator Learning under Partial Observations

Neural operators learn mappings between function spaces, but are typically developed with dense input-output training fields and fully observed inputs at inference. Many scientific problems require instead predicting solution fields from sparse, irregular, or partial observations under uncertainty. We introduce Neural Operator Processes (NOPs), a framework that unifies neural-process conditioning with neural-operator decoding to predict full output fields from limited context. NOPs condition on sparse joint input-output observations and support deterministic and probabilistic prediction within a shared encoder-decoder architecture. We study two conditioning strategies, convolutional pooled summaries and query-aligned attention, and analyze how their interaction with latent stochastic variables depends on PDE geometry. Across function regression and three PDE benchmarks, we find that sparse conditional operator learning is viable and can match dense-grid behavior in several regimes, that preserving local context-query geometry is essential in non-periodic settings but less so in spectrally smooth periodic regimes, and that uncertainty-aware operator learning succeeds when latent conditioning complements rather than overwrites the local geometric pathway. These results provide a basis for probabilistic operator learning under partial observations and help bridge operator learning and probabilistic meta-learning in function space.

Jose Miguel Lara-Rangel, Serge Guillas

University College London London, United Kingdom

Neural ProcessesPDE Operator LearningOperator LearningNeural Operators
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