Deep Operator Networks

Also known as DeepONet

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1 paper in the last four weeks, against 2 the four weeks before. 0.0% of all new papers.

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Latest papers 27

Sep 28, 2026cs.CV

Spectral Super-Resolution using Spatial-Spectral Residual Operator Networks

Spectral super-resolution of multispectral satellite images can enable high temporal- and spatial-resolution hyperspectral satellite imagery at a modest cost, significantly increasing the applicability of hyperspectral remote sensing. This task is inherently ill-posed, making it well-suited for deep learning-based methods. In this study, the spectral super-resolution task is framed as an operator learning problem, and SSRON is proposed as a Deep Operator Network that effectively learns function-to-function mappings from downsampled spectra to continuous spectra. The model is trained to super-resolve Sentinel-2A-like multispectral imagery to EMIT images. Compared to baseline models, SSRON achieves superior performance across all metrics. The model also demonstrates zero-shot spectral super-resolution capability by predicting bands unseen during training. Furthermore, its continuous-output formulation suggests the potential to estimate spectra at finer wavelength intervals than the native sensor. These results suggest the potential of SSRON and establishes operator learning as a promising direction for spectral super-resolution.

Seokhyun Chin

California Institute of Technology 1200 E California Boulevard, Pasadena, CA, USA

Hyperspectral Image Super-Resolution Hyperspectral Imaging Deep Operator Networks

Sep 1, 2026cs.LG

GenONet: A Generative operator Network for High-Resolution Precipitation Nowcasting

High-resolution precipitation nowcasting is critical for reducing the impacts of severe weather but remains difficult because of rapid storm evolution. Deep learning models have shown great promise for this task, but their predictive skill often deteriorates over longer forecast horizons. This leads to increasingly blurry forecasts that fail to capture the complex, non-linear evolution of storm systems. In order to address these limitations, we introduce Spatio-Temporal U-DeepONet (GenONet), a novel architecture for long-range precipitation forecasting up to 3 hours, specifically designed to produce sharp and physically consistent results. GenONet's architecture pioneers the use of a Deep Operator Network (DeepONet) as a generator within a Generative Adversarial Network (GAN) framework for this task. The DeepONet learns the continuous-time dynamics of precipitation, ensuring stability over long forecast horizons. Adversial training against a spatio-temporal discriminator compels the model to produce sharp, coherent forecasts, while a physics-informed loss regularizer, derived from the Moisture Conservation Equation, improves physical plausibility in our ablation setting. Quantitative evaluations show that our model achieves consistently higher scores on most of the metrics, especially for highintensity events and at longer lead times. Qualitatively, GenONet produces structurally coherent forecasts that maintain their integrity, whereas baseline models degrade into indistinct patterns. Finally, an ablation study confirms the benefit of this physics-informed loss, highlighting the strength of combining operator learning with adversarial training.

Mohammad Kian Golkar, Luciano Alves de Oliveira, Mohammad Khanjani

University of Illinois Urbana-Champaign · Sharif University of Technology

Precipitation Forecasting Spatiotemporal Forecasting Deep Operator Networks Physics-Informed Neural Operators Precipitation Nowcasting Generative Adversarial Networks

Aug 10, 2026math.NA

Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fields

Forward uncertainty propagation in complex physical systems can induce structured covariance across field-valued outputs. For a probabilistic surrogate, the total predictive covariance comprises the covariance of conditional means across input realizations and the average conditional predictive covariance. Probabilistic DeepONet (Prob-DeepONet) provides lightweight uncertainty quantification by predicting pointwise Gaussian means and variances in a single forward pass, but its conditional predictive covariance is restricted to a diagonal form. To represent cross-location conditional dependence without explicitly parameterizing a full high-dimensional covariance matrix, we develop a two-step mean-variance DeepONet (two-step MV-DeepONet) through two principal modifications. First, two-step training is used to decouple output-basis learning from the input-to-coefficient mapping, together with basis orthogonalization and subspace rotation. Second, Gaussian probabilistic modeling is transferred from the high-dimensional physical output space to the low-dimensional rotated coefficient space. Mapping these probabilistic coefficients through the shared basis induces a generally non-diagonal conditional predictive covariance in the physical output space while retaining single-pass inference. A Frobenius-norm error decomposition and corresponding upper bound identify low-rank covariance compressibility, trunk-subspace approximation, finite-sample statistical error, and coefficient-space covariance estimation as the principal factors governing covariance recovery. Numerical experiments on three representative problems governed by partial differential equations (PDEs) and a hypersonic blunt-body aerothermal problem show improved generalization, more structured uncertainty bands, and accurate recovery of off-diagonal correlation patterns compared with Prob-DeepONet.

Yupei Nie, Lei Wang, Jiasen Liu

School of Mathematics and Physics, North China Electric Power University, Beijing 102206, P.R. China · School of Energy Power and Mechanical Engineering, North China Electric Power University, Beijing 102206, P.R. China

Uncertainty Quantification PDE Surrogate Modeling Deep Operator Networks Operator Learning Uncertainty Propagation

Aug 5, 2026cs.LG

Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spaces

Deep Operator Networks (DeepONets; arXiv:1910.03193) typically encode an input function through point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov (arXiv:2603.11972), we replace point samples by continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space

({V},\{p_α\}_{α\in A})

, whose topology is generated by a point-separating family of seminorms rather than a single norm, and develop fixed and adaptive functional measurement systems. Measurements are combined with the coefficient-space Two-Step procedure of Lee and Shin (arXiv:2309.01020), while a training-only decoder and regularization stabilize the adaptive coordinates. We derive a discrete error decomposition separating measurement, output-basis, and neural-approximation errors, together with a Barron-rate refinement. The framework is evaluated on the antiderivative operator, a non-normable locally convex input space, heterogeneous Darcy flow, a controlled operator, and fixed-time and time-evolving Navier-Stokes vorticity operators. In the heterogeneous Darcy problem, the functional models retain nearly resolution-independent errors of 5.5-5.6% on unseen grids, while in the controlled problem adaptive measurements reduce the mean error below 1.2%. For the fixed-time Navier-Stokes problem, the Adaptive Topological DeepONet is the most accurate DeepONet-based model, attaining a mean relative

L^2

error of 1.685% +/- 0.017% using 128 functional coordinates. A comparably sized Fourier neural operator (FNO; arXiv:2010.08895) achieves the lower error 0.832% +/- 0.172%, but requires the full 64x64 input field, twice the training time, and 10.7x greater peak GPU memory. The formulation provides compact, interpretable, and discretization-portable coordinates in the continuous dual

V'

, including for non-normable input spaces.

Khemraj Shukla, George Em Karniadakis

Division of Applied Mathematics, Brown University Providence, Rhode Island 02912, USA

PDE Operator Learning Neural Network Approximation Theory Deep Operator Networks

Jul 30, 2026cs.LG

Feature Interaction Modeling for Neural Operators

Despite the many variants of DeepONet that have been proposed, query-based operator networks still struggle with shock-dominated and low-viscosity PDEs, whose sharp moving discontinuities and slowly decaying solution spectra challenge finite-dimensional separable representations. In this work, we propose \emph{Feature Interaction Modeling Operator} (FM-Operator), a point-wise query neural operator that explicitly models feature construction and interactions between sensor observations and query coordinates. Our design is motivated by a reinterpretation of the canonical DeepONet aggregation through the lens of multiplicative interactions. Specifically, the branch--trunk inner product admits the equivalent form

\boldsymbol{b}(u)^\top \boldsymbolτ(y)=\boldsymbol{1}^\top \operatorname{diag}(\boldsymbol{b}(u))\,\boldsymbolτ(y)

, revealing that the two representations interact only along corresponding latent dimensions and therefore constitute a diagonally constrained multiplicative interaction. This observation suggests that, beyond improving the individual branch and trunk networks, the structure through which function and query representations interact is itself an important inductive bias in point-wise operator learning. FM-Operator accordingly redesigns both feature construction and feature interaction, enabling structured information exchange beyond the conventional branch--trunk coupling while retaining point-wise query evaluation. Experiments across multiple PDE benchmarks demonstrate that FM-Operator consistently outperforms vanilla DeepONet and achieves clear improvements over the strong Shift-DeepONet baseline. These results suggest that explicitly designing representation construction and interaction provides a promising direction for improving the effectiveness of DeepONet-style query-based neural operators.

Quan Gu, Xiaoduo Li, Hongxia Liu

Independent Researcher · Taiyuan University of Technology

Feature Interaction Modeling PDE Surrogate Modeling PDE Operator Learning Deep Operator Networks

Jul 29, 2026cs.LG

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,

L^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

L^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

L^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

School of Computer Science Georgia Institute of Technology USA

Quantitative Finance Deep Operator Networks Density Estimation Neural Operators Inverse Problems Finance

Jul 28, 2026cs.LG

SpectONet: A Physics-Guided Spectral Deep Operator Network for Euler-Bernoulli Beam Dynamics

This paper proposes a novel physics-guided spectral deep operator network, termed SpectONet, for solving Euler-Bernoulli beam (EBB) vibration problems. The proposed framework integrates the operator-learning capability of DeepONet with physics-informed constraints and Chebyshev-Gauss-Lobatto (CGL) sensor placement. Unlike conventional DeepONet frameworks, which commonly employ uniformly distributed sensors, SpectONet uses nonuniform spectral sensor locations with a higher concentration of points near the domain boundaries. This sampling strategy improves the finite-dimensional representation of boundary-sensitive structural responses while requiring only a limited number of branch-network inputs. The governing beam equation, together with the associated initial and boundary conditions, incorporated into the training objective to promote physically consistent and generalizable predictions. Numerical experiments on three synthetic EBB vibration problems and a real-world bridge vibration dataset demonstrate the effectiveness of the proposed framework. Comparisons with strong baselines such as, Vanilla DeepONet, PI-DeepONet, PINN, and CNN-UNet show that SpectONet consistently achieves lower prediction errors across all considered evaluation metrics. In particular, SpectONet achieves at least

64\%

improvement over the considered baseline models across the three synthetic problems and at least

37\%

for the real-world problems. These results demonstrate that SpectONet provides an accurate, computationally efficient, and physically consistent operator-learning framework for structural vibration analysis.

Shivani Saini, Ramesh Kumar Vats, Arup Kumar Sahoo

Department of Mathematics and Scientific Computing, National Institute of Technology Hamirpur, 177005, Himachal Pradesh, India · The Hatter Department of Marine Technologies, Leon H. Charney School of Marine Sciences, University of Haifa, 3498838, Haifa, Israel

Structural Mechanics PDE Operator Learning Deep Operator Networks Physics-Informed Neural Operators

Jul 10, 2026cs.LG

Learning Physics-Informed Surrogate Model of Linear Elastic Displacement Fields from Geometry

This work aims to develop a fast and physically consistent surrogate model for real-time structural health monitoring of fractured elastic domains. We propose a physics-informed DeepONet framework that predicts displacement fields from both boundary conditions and fracture geometry, using a dedicated encoding strategy for the latter and without relying on finite-element-generated training data. The traction-free condition on the fracture boundary is imposed weakly through a localized penalty term. The presented numerical example focuses on one representative fracture geometry, demonstrating the feasibility of the formulation and laying the groundwork for extensions to surrogate modeling across diverse fracture geometries.

Rodolphe Barlogis, Ferhat Tamssaouet, Quentin Falcoz +1

PROMES-CNRS, Universit´e Perpignan Via Domitia (UPVD), Perpignan, France · LAAS-CNRS, Universit´e de Toulouse (UT), Toulouse, France

Structural Mechanics PDE Surrogate Modeling Deep Operator Networks Physics-Informed Neural Operators Structural Health Monitoring

Jul 8, 2026cs.LG

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

L^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 Descent Neural Network Optimization Deep Operator Networks Operator Learning Neural Operators

Jun 18, 2026cs.LG

Adaptive Distance-Aware Trunk Deep Operator Learning for Long-Span Roadway Bridges

Long-span roadway bridges exhibit highly localized structural responses under vehicular loading, making repeated FE analysis computationally expensive for applications such as influence surface generation and structural digital twins. Existing SciML approaches struggle to accurately capture these localized responses. To address this challenge, this study proposes an adaptive-trunk DeepONet for localized structural response prediction in large-scale bridge systems. The framework dynamically constructs a load-dependent learning domain using a KNN strategy, allowing the network to focus on structural influence zones. The trunk network is further enhanced using distance-aware features that encode the geometric relationship between the load and structural nodes. A physics-based full-field reconstruction is incorporated through a stiffness-informed Schur complement formulation, enabling predictions at adaptive nodes to be extended to the entire structural domain. To enable scalable training, response data are generated using a reduced-order equivalent shell model that preserves the dominant global behavior while significantly reducing computational cost. The proposed framework is validated on both a benchmark bridge model and the real-world Mussafah Bridge. Results show that the method achieves FEM-level accuracy with relative errors below 5%, while reducing the total response evaluation time (including full-field reconstruction) by approximately 60x; excluding the post-processing reconstruction step, the AD-DeepONet inference is up to four orders of magnitude faster than FEM. In addition, the framework enables rapid generation of full-field responses, influence lines, and influence surfaces under arbitrary vehicular loading configurations, demonstrating strong potential for large-scale bridge analysis and digital twin applications.

Bilal Ahmed, Diab W. Abueidda, Waleed El-Sekelly +2

Civil and Urban Engineering Department, New York University Abu Dhabi, United Arab Emirates · New York University Abu Dhabi, United Arab Emirates · Department of Structural Engineering, Mansoura University, Mansoura, Egypt

Deep Operator Networks

Jun 16, 2026cs.LG

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 Modeling Deep Operator Networks Neural Network Compression Neural Operators

May 19, 2026cs.LG

Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions

Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing approaches typically approximate such features within continuous function spaces, often requiring increased model capacity and high-resolution data. In this work, we propose Cut-DeepONet, a two-stage training framework that explicitly models discontinuities while reducing learning complexity. Our approach reformulates the problem via a lifting strategy, partitioning the domain into smooth subregions while representing discontinuities as boundaries in a higher-dimensional space. This separation aligns the operator learning task with the inductive bias of neural networks and avoids directly approximating discontinuities. An additional network predicts input-dependent discontinuity locations for unseen inputs, which are then used to guide the neural operator in generating smooth components within each region. Experiments on benchmark PDEs show that Cut-DeepONet outperforms state-of-the-art methods, even when trained on low-resolution datasets. The method excels on problems with discontinuities and sharp transitions, while using fewer trainable parameters. Our results highlight the benefits of changing the representation of operator learning rather than increasing model complexity.

Ha Dang, Sebastian Schmidt, Juergen Hesser

Mannheim Institute for Intelligent Systems in Medicine, Heidelberg University · Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University · Heidelberg Institute for Theoretical Studies (HITS), Heidelberg University +2

PDE Operator Learning Deep Operator Networks Operator Learning

May 18, 2026cs.RO

Neural Operators for Design-Space Surrogate Modeling of Tendon-Actuated Continuum Robots

Continuum robots enable dexterous manipulation in constrained environments, but require accurate and efficient models for real-time manipulation and control. Traditional physics-based models can be computationally expensive and may suffer from inaccuracies due to unmodeled effects, while current learning-based methods often generalize poorly beyond the specific robot on which they are trained. We present a formulation of surrogate modeling for tendon-driven continuum robots as an operator learning problem that maps robot design parameters and tendon actuation inputs to resulting configurations. This formulation enables a single trained model to generalize across a large class of robot designs. We develop four novel neural operator architectures--two based on Deep Operator Networks (DeepONets) and two based on Fourier Neural Operators (FNOs)--and train them on simulation data to predict robot configurations. All architectures achieve good accuracy while allowing for fast and accurate generalization across designs. Our results demonstrate that operator learning provides an effective and generalizable surrogate for continuum robot mechanics in the design space, enabling fast modeling for control, planning, and design optimization in surgical and industrial applications.

Branden Frieden, James M. Ferguson, Alan Kuntz +1

The Robotics Center and the Kahlert School of Computing at the University of Utah, Salt Lake City, UT 84112, USA. · The Departments of Computer Science and Electrical and Computer Engineering at Vanderbilt University, Nashville, TN, 37203, USA.

Neural Surrogate Modeling Fourier Neural Operator Continuum Robot Control Tendon-Driven Actuation Deep Operator Networks Neural Operators

May 16, 2026stat.ML

Diffusion-Based Stochastic Operator Networks for Uncertainty Quantification in Stochastic Partial Differential Equations

We introduce a novel framework for uncertainty quantification of solution operators associated with stochastic partial differential equations (SPDEs). Although SPDEs play a central role in modeling complex physical systems under uncertainty, their practical use typically requires specifying the magnitude and structure of model uncertainties that are often unknown and difficult to infer from noisy measurements. To address this challenge, we develop a stochastic operator-learning framework that learns directly from noisy data and outputs both a mean solution field and a quantification of uncertainty. The proposed method, namely the Stochastic Operator Network (SON), is constructed by combining the structure of the Deep Operator Network (DeepONet) with Stochastic Neural Networks (SNNs) to model stochasticity and enable probabilistic prediction. The training procedure is carried out by minimizing a Hamiltonian-type loss and optimizing the resulting objective using the Stochastic Maximum Principle. Numerical experiments on benchmark SPDEs under multiple uncertainty sources demonstrate the accuracy and robustness of the proposed method in capturing solution structure and quantifying predictive uncertainty.

Phuoc-Toan Huynh, Richard Archibald, Feng Bao

Department of Mathematics, Florida State University, Tallahassee, FL, USA · Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA

Uncertainty Quantification Deep Operator Networks Operator Learning Neural Operators

May 15, 2026math.NA

fPINN-DeepONet: A Physics-Informed Operator Learning Framework for Multi-term Time-fractional Mixed Diffusion-wave Equations

In this paper, we develop a physics-informed deep operator learning framework for solving multi-term time-fractional mixed diffusion-wave equations (TFMDWEs). We begin by deriving an

L_2

approximation, which achieves first-order accuracy for the Caputo fractional derivative of order

β\in (1,2)

. Building upon this foundation, we propose the fPINN-DeepONet framework, a novel approach that integrates operator learning with the

L_2

approximation to efficiently solve fractional partial differential equations (FPDEs). Our framework is successfully applied to both fixed and variable fractional-order PDEs, demonstrating the framework's versatility and broad applicability. To evaluate the performance of the proposed model, we conduct a series of numerical experiments that involve dynamically varying fractional orders in both space and time, as well as scenarios with noisy data. These results highlight the accuracy, robustness, and efficiency of the fPINN-DeepONet framework.

Binghang Lu, Zhaopeng Hao, Christian Moya +1

Department of Mathematics, Purdue University West Lafayette, IN 47907, USA · School of Mechanical Engineering, Purdue University West Lafayette, IN 47907, USA

PDE Solving Deep Operator Networks Physics-Informed Neural Operators

May 15, 2026stat.ML

A numerical study into neural network surrogate model performance for uncertainty propagation

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

Noah Wade, Kirubel Teferra

ASEE Postdoctoral Associate, U.S. Naval Research Laboratory, Washington, DC, 20375, USA · U.S. Naval Research Laboratory, Washington, 20375, DC, USA

Neural Surrogate Modeling Uncertainty Quantification PDE Surrogate Modeling OOD Generalization Deep Operator Networks Uncertainty Propagation

May 13, 2026cs.LG

Frequency Bias and OOD Generalization in Neural Operators under a Variable-Coefficient Wave Equation

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

Runlong Xie, An Luo

Independent Researcher · School of Statistics, University of Minnesota, MN, USA.

Fourier Neural Operator PDE Solving OOD Generalization Spectral Bias PDE Operator Learning Deep Operator Networks

May 8, 2026math.NA

NSPOD: Accelerating Krylov solvers via DeepONet-learned POD subspaces

The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, the location/values of the applied Dirichlet/Neumann boundary conditions, body forces and material properties, among others. We have previously introduced hybridization of classical linear iterative solvers with neural operators for specific geometries, but they tend to not perform well on geometries not previously seen during training. We partially addressed this challenge by introducing the deep operator network Geo-DeepONet and hybridizing it with Krylov-based iterative linear solvers, which, despite learning effectively across arbitrary unstructured meshes without requiring retraining, led to only modest reductions in iterations compared to state-of-the-art preconditioners. In this study we introduce Neural Subspace Proper Orthogonal Decomposition (NSPOD), a multigrid-like deep operator network-based preconditioner which can dramatically reduce the number of iterations needed for convergence in Krylov-based linear iterative solvers, even when compared to state-of-the-art methods such as algebraic multigrid preconditioners. We demonstrate its efficiency via numerical experiments on a linearized version of solid mechanics PDEs applied to unstructured domains obtained from complex CAD geometries. We expect that the findings in this study lead to more efficient hybrid preconditioners that can match, or possibly even surpass, the convergence properties of the current gold standard preconditioning methods for solid mechanics PDEs.

Francesc Levrero-Florencio, Youngkyu Lee, Jay Pathak +1

Synopsys UK Ltd, UK · Division of Applied Mathematics, Brown University, Rhode Island, USA · Synopsys Inc, California, USA

PDE Solving Structural Mechanics Proper Orthogonal Decomposition Deep Operator Networks Krylov Subspace Methods

May 1, 2026cs.LG

Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometries

This paper deals with solving the 2D Helmholtz equation on non-parametric domains, leveraging a physics-informed neural operator network based on the DeepONet framework. We consider a 2D square domain with an inclusion of arbitrary boundary geometry at its center. This inclusion acts as a scatterer for an incoming harmonic wave. The aim is to learn the operator linking the geometry of the scatterer to the resulting scattered field. A signed distance function to the boundary of the inner inclusion, evaluated at several points in the domain, is used to encode its geometry. It serves as input for the branch part of the DeepONet architecture, while local information is used as input for the trunk part. This approach enables the encoding of arbitrary geometries, whether they are parameterized or not. The evaluation of the model on unseen geometries is compared with its finite element method (FEM) equivalent to test its generalization capabilities. The trained network weights implicitly embed the local physics and their interaction with the domain geometry. If the training space sufficiently covers the target evaluation space, the model can generalize accordingly. Furthermore, it can be refined to extend to another region of interest without retraining from scratch. This framework also avoids the need to remesh the domain for each geometry. The proposed approach delivers a computationally lighter surrogate model than FEM alternatives and avoids relying on FEM-generated training data.

Rodolphe Barlogis, Ferhat Tamssaouet, Quentin Falcoz +1

PROMES-CNRS, Université Perpignan Via Domitia (UPVD), Perpignan, France · LAAS-CNRS, Université de Toulouse (UT), Toulouse, France

PDE Solving PDE Operator Learning Deep Operator Networks Physics-Informed Neural Operators

May 1, 2026cs.LG

Conformalized Quantum DeepONet Ensembles: Towards Scalable Operator Learning with Distribution-Free Guarantees

Operator learning enables fast surrogate modelling of high-dimensional dynamical systems, but existing approaches face two fundamental limitations: the quadratic cost of dense neural layers and unreliable uncertainty quantification in safety-critical settings. We propose Conformalized Quantum DeepONet Ensembles, a framework that addresses both challenges simultaneously. Using Quantum Orthogonal Neural Networks (QOrthoNNs), we characterize the resource regime in which their established

\widetilde{\mathcal{O}}(n)

hidden-layer running-time scaling improves on the

\mathcal{O}(n^2)

cost of a classical dense layer. To quantify uncertainty, we combine ensemble predictions with split conformal calibration. For a new input-output function pair jointly exchangeable with the calibration pairs, we prove a finite-sample, distribution-free lower bound on the expected fraction of covered query locations. As a secondary proof-of-concept, we explore hybrid classical-quantum architectures and superposed execution to manage ensemble runtime and hardware requirements. Experiments on synthetic operator benchmarks and real-world power-system dynamics show accurate predictions under ideal simulation and empirical coverage near the target under both ideal conditions and selected compact-circuit simulations with depolarizing or device-calibrated composite noise. Together, these results connect resource-aware scalability analysis with finite-sample uncertainty guarantees for quantum operator learning.

Purav Matlia, Christian Moya, Guang Lin

Department of Computer Science Purdue University · Department of Mathematics Purdue University · Department of Mathematics Department of Mechanical Engineering Purdue University

Uncertainty Quantification Hybrid Quantum-Classical ML Deep Operator Networks Quantum Neural Networks

Apr 21, 2026cs.LG

Physics-Guided Dimension Reduction for Simulation-Free Operator Learning of Stiff Differential-Algebraic Systems

Neural surrogates for stiff differential-algebraic equations (DAEs) face two barriers: soft-constraint methods leave algebraic residuals that stiffness amplifies into errors, and hard-constraint methods require trajectory data from stiff integrators. We introduce an extended Newton implicit layer that enforces algebraic constraints exactly and reduces fast dynamics to their quasi-steady-state values in a single differentiable solve. Embedded in a physics-informed DeepONet, the layer recovers all fast and algebraic states exactly from slow-state predictions, removes the per-window stiffness-amplification pathway, and yields a stiffness-scaled Implicit Function Theorem gradient absent from penalty methods. Cascaded implicit layers extend this to multi-component systems with provable convergence. On a grid-forming inverter (stiffness ratio of about 4712), extended Newton attains 1.42% error versus 39.3% (penalty) and 57.0% (standard Newton); augmented Lagrangian and feedback linearization diverged. Two independently trained models compose without retraining (0.72% to 1.16% error, exact constraint satisfaction). Cross-domain validation on the Robertson stiff DAE (stiffness ratio up to

10^5

) confirms generalization. Conformal prediction provides 90% coverage with automatic out-of-distribution detection.

Huy Hoang Le, Haoguang Wang, Christian Moya +2

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 USA · Department of Mathematics, Purdue University, West Lafayette, IN 47907 USA · Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07102 USA

Reduced-Order Modeling Deep Operator Networks Operator Learning

Apr 20, 2026math.OC

Learning the Riccati solution operator for time-varying LQR via Deep Operator Networks

We propose a computational framework for replacing the repeated numerical solution of differential Riccati equations in finite-horizon Linear Quadratic Regulator (LQR) problems by a learned operator surrogate. Instead of solving a nonlinear matrix-valued differential equation for each new system instance, we construct offline an approximation of the associated solution operator mapping time-dependent system parameters to the Riccati trajectory. The resulting model enables fast online evaluation of approximate optimal feedbacks across a wide class of systems, thereby shifting the computational burden from repeated numerical integration to a one-time learning stage. From a theoretical perspective, we establish control-theoretic guarantees for this operator-based approximation. In particular, we derive bounds quantifying how operator approximation errors propagate to feedback performance, trajectory accuracy, and cost suboptimality, and we prove that exponential stability of the closed-loop system is preserved under sufficiently accurate operator approximation. These results provide a framework to assess the reliability of data-driven approximations in optimal control. On the computational side, we design tailored DeepONet architectures for matrix-valued, time-dependent problems and introduce a progressive learning strategy to address scalability with respect to the system dimension. Numerical experiments on both time-invariant and time-varying LQR problems demonstrate that the proposed approach achieves high accuracy and strong generalization across a wide range of system configurations, while delivering substantial computational speedups compared to classical solvers. The method offers an effective and scalable alternative for parametric and real-time optimal control applications.

Jun Chen, Umberto Biccari, Junmin Wang

Feedback Control Deep Operator Networks Operator Learning Linear Quadratic Regulator Optimal Control

Mar 18, 2026cs.LG

Translation Invariance of Neural Operators for the FitzHugh-Nagumo Model

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

Luca Pellegrini

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

Fourier Neural Operator PDE Operator Learning Deep Operator Networks Neural Operators

Dec 9, 2025cs.LG

Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation

Deep Operator Networks (DeepONets) have emerged as a powerful framework for data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk network, which operates directly on raw spatial or spatiotemporal coordinates through fully connected layers, often struggles to represent sharp gradients, boundary layers, and other non-periodic solution structures on bounded domains. To address these limitations, we introduce the Spectral-Embedded Deep Operator Network (SEDONet), a novel DeepONet architecture in which the trunk is driven by a fixed Chebyshev spectral dictionary instead of coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias for bounded domains, enabling the learned operator to capture fine-scale features that are difficult for Fourier-based or MLP-only trunks to represent. SEDONet is evaluated on the 2-D Poisson equation, 1-D Burgers' equation, 1-D advection-diffusion equation, Allen-Cahn equation, Lorenz-96 chaotic system, and Darcy flow, covering elliptic, hyperbolic, parabolic, chaotic, and multiscale problems. Across all benchmarks, SEDONet consistently achieves the lowest or statistically comparable relative

L^2

errors among DeepONet, FEDONet, and SEDONet, with improvements of up to 54% over the baseline DeepONet and consistent gains over Fourier-embedded variants on bounded, non-periodic problems. Energy spectrum analyses further demonstrate that SEDONet more accurately preserves intermediate- and high-frequency solution structures. The proposed framework provides a simple, parameter-neutral modification to DeepONets, offering a robust and computationally efficient spectral approach for surrogate modeling of nonlinear operators in scientific computing.

Muhammad Abid, Omer San

Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN 37996, USA.

PDE Surrogate Modeling PDE Operator Learning Spectral Methods Deep Operator Networks

Mar 31, 2025math.FA

New universal operator approximation theorem for encoder-decoder architectures

Motivated by the rapidly growing field of mathematics for operator approximation with neural networks, we present a novel universal operator approximation theorem for broad classes of encoder-decoder architectures and a wide range of input and output spaces. In this study, we focus on the approximation of continuous operators between infinite-dimensional normed or metric spaces in the topology of uniform convergence on compact sets. Unlike standard results in the operator learning literature, we additionally investigate the case where the approximating sequence of encoder-decoder architectures can be chosen independently of the compact sets. Taking a topological perspective, we point out that compact-set-independent approximation is a strictly stronger property in most relevant operator learning frameworks. To establish our results, we introduce new approximation properties of input and output spaces tailored to encoder-decoder architectures. These properties enable us to prove a universal operator approximation theorem ensuring uniform convergence on every compact subset of the input space. Our results unify and extend existing universal operator approximation theorems for various encoder-decoder architectures, including classical DeepONets, BasisONets, MIONets, architectures based on frames and other related approaches. A notable feature of our framework is that it also applies to metric spaces beyond the normed setting. In particular, it allows the consideration of

p

-Wasserstein spaces of probability measures as input or output spaces, and Skorohod spaces of càdlàg functions as input spaces. This generality also opens up potential applications in optimal transport.

Janek Gödeke, Pascal Fernsel

Center for Industrial Mathematics University of Bremen

Neural Network Approximation Theory Deep Operator Networks Operator Learning

Mar 7, 2025cs.CE

From Theory to Application: A Practical Introduction to Neural Operators in Scientific Computing

This review examines neural operator architectures for learning solution operators of parametric partial differential equations (PDEs), with an emphasis on conceptual clarity and practical implementation. The work analyzes key models, including DeepONet, PCANet, and the Fourier Neural Operator, highlighting their underlying representations, computational structures, and comparative performance. These architectures are demonstrated on three canonical PDE problems: the Poisson equation, a linear elasticity problem, and a hyperelasticity problem. To make the presentation self-contained, key foundational topics are introduced, including finite-dimensional representations of function spaces, singular-value decomposition, and sampling from infinite-dimensional function spaces. Beyond forward modeling, the review discusses the use of neural operators as surrogate models within a Bayesian inverse-problem framework, including prior specification, forward-map approximation, and posterior computation. The performance of the three neural-operator architectures is evaluated on in-distribution samples, out-of-distribution samples, and Bayesian inference tasks. The review also discusses challenges related to prediction accuracy and generalization, outlining emerging strategies such as residual-based error correction and multi-level training. The review concludes by positioning neural operators within broader scientific-computing workflows and by identifying directions for reliable, scalable operator learning.

Prashant K. Jha

Department of Mechanical Engineering, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA.

Neural Surrogate Modeling Fourier Neural Operator PDE Operator Learning Deep Operator Networks Neural Operators

Dec 15, 2021math.NA

Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations

We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial differential equations. In particular, we consider problems set in

d

-dimensional periodic domains,

d=1, 2, \dots

, and with analytic right-hand sides and coefficients. Our analysis covers linear, elliptic second order divergence-form PDEs as, e.g., diffusion-reaction problems, parametric diffusion equations, and elliptic systems such as linear isotropic elastostatics in heterogeneous materials. We leverage the exponential convergence of spectral collocation methods for boundary value problems whose solutions are analytic. In the present periodic and analytic setting, this follows from classical elliptic regularity. Within the ONet branch and trunk construction of [Chen and Chen, 1993] and of [Lu et al., 2021], we show the existence of deep ONets which emulate the coefficient-to-solution map to a desired accuracy in the

H^1

norm, uniformly over the coefficient set. We prove that the neural networks in the ONet have size

\mathcal{O}(\left|\log(\varepsilon)\right|^κ)

, where

\varepsilon>0

is the approximation accuracy, for some

κ>0

depending on the physical space dimension.

Carlo Marcati, Christoph Schwab

Seminar for Applied Mathematics, ETH Zurich, Switzerland

PDE Solving PDE Operator Learning Neural Network Approximation Theory Deep Operator Networks