Deep Operator Networks

Also known as DeepONet

Ask about this topic

Overview

Deep Operator Networks (DeepONets) learn operators that map an input function, such as a PDE coefficient or forcing term, to a solution function, typically using a branch network for input measurements and a trunk network for output coordinates. They are used to make repeated simulations faster in settings including fluid flow, materials, waves, and control, with approaches ranging from physics-informed training to spectral embeddings, attention, and localized expert models. Theory provides universal approximation results, while practical evaluations commonly use PDE benchmarks and compare accuracy, inference cost, and generalization across unseen inputs or geometries. A central challenge is that strong performance on familiar data may not transfer to high-frequency inputs, sharp transitions, or new geometries, and accuracy must be balanced against data needs, computational cost, and uncertainty estimates.

Latest papers

CardsList

51 papers4 papers in the last four weeks, level with the four weeks before. 0.0% of all new papers.

May 8, 2026

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 SolvingStructural MechanicsProper Orthogonal DecompositionDeep Operator NetworksKrylov Subspace Methods
AskPDF
May 1, 2026

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 SolvingPDE Operator LearningDeep Operator NetworksPhysics-Informed Neural Operators
AskPDF
May 1, 2026

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 O~(n)\widetilde{\mathcal{O}}(n) hidden-layer running-time scaling improves on the O(n2)\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 QuantificationHybrid Quantum-Classical MLDeep Operator NetworksQuantum Neural Networks
AskPDF
Apr 21, 2026

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 10510^5) 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 ModelingDeep Operator NetworksOperator Learning
AskPDF
Apr 20, 2026

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 ControlDeep Operator NetworksOperator LearningLinear Quadratic RegulatorOptimal Control
AskPDF
Apr 13, 2026

Multi-Head Residual-Gated DeepONet for Coherent Nonlinear Wave Dynamics

Coherent nonlinear wave dynamics are often strongly shaped by a compact set of physically meaningful descriptors of the initial state. Traditional neural operators typically treat the input-output mapping as a largely black-box high-dimensional regression problem, without explicitly exploiting this structured physical context. Common feature-integration strategies usually rely on direct concatenation or FiLM-style affine modulation in hidden latent spaces. Here we introduce a different paradigm, loosely inspired by the complementary roles of state evolution and physically meaningful observables in quantum mechanics: the wave field is learned through a standard DeepONet state pathway, while compact physical descriptors follow a parallel conditioning pathway and act as residual modulation factors on the state prediction. Based on this idea, we develop a Multi-Head Residual-Gated DeepONet (MH-RG), which combines a pre-branch residual modulator, a branch residual gate, and a trunk residual gate with a low-rank multi-head mechanism to capture multiple complementary conditioned response patterns without prohibitive parameter growth. We evaluate the framework on representative benchmarks including highly nonlinear conservative wave dynamics and dissipative trapped dynamics and further perform detailed mechanistic analyses of the learned multi-head gating behavior. Compared with feature-augmented baselines, MH-RG DeepONet achieves consistently lower error while better preserving phase coherence and the fidelity of physically relevant dynamical quantities.

Zhiwei Fan, Yiming Pan, Daniel Coca

School of Engineering, Newcastle University, NE1 7RU, United Kingdom · School of Physical Science and Technology, ShanghaiTech University, Shanghai 200031, China

Deep Operator NetworksNeural OperatorsDynamical SystemsResidual Learning
AskPDF
Apr 7, 2026

Operator Learning for Predicting Bulk Wave Parameters of Spectral Wave Models

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

Shukai Cai, Sourav Dutta, Mark Loveland +4

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

PDE Surrogate ModelingPDE Operator LearningDeep Operator Networks
AskPDF
Mar 23, 2026

CoNBONet: Conformalized Neuroscience-inspired Bayesian Operator Network for Reliability Analysis

Time-dependent reliability analysis of nonlinear dynamical systems under stochastic excitations is a critical yet computationally demanding task. Conventional approaches, such as Monte Carlo simulation, necessitate repeated evaluations of computationally expensive numerical solvers, leading to significant computational bottlenecks. To address this challenge, we propose CoNBONet, a neuroscience-inspired surrogate model that enables fast, energy-efficient, and uncertainty-aware reliability analysis, providing a scalable alternative to techniques such as Monte Carlo simulations. CoNBONet, short for Conformalized Neuroscience-inspired Bayesian Operator Network, leverages the expressive power of deep operator networks while integrating neuroscience-inspired neuron models to achieve fast, low-power inference. Unlike traditional surrogates such as Gaussian processes, polynomial chaos expansions, or support vector regression, that may face scalability challenges for high-dimensional, time-dependent reliability problems, CoNBONet offers fast and energy-efficient inference enabled by a neuroscience-inspired network architecture, calibrated uncertainty quantification with theoretical guarantees via split conformal prediction, and strong generalization capability through an operator-learning paradigm that maps input functions to system response trajectories. Validation of the proposed CoNBONet for various nonlinear dynamical systems demonstrates that CoNBONet preserves predictive fidelity, and achieves reliable coverage of failure probabilities, making it a powerful tool for robust and scalable reliability analysis in engineering design.

Shailesh Garg, Souvik Chakraborty

Department of Applied Mechanics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India. · Department of Applied Mechanics, Yardi School of Artificial Intelligence (YScAI), Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India.

Uncertainty QuantificationBayesian Neural NetworksDeep Operator Networks
AskPDF
Mar 22, 2026

Direct Interval Propagation Methods using Neural-Network Surrogates for Uncertainty Quantification in Physical Systems Surrogate Model

In engineering, uncertainty propagation aims to characterise system outputs under uncertain inputs. For interval uncertainty, the goal is to determine output bounds given interval-valued inputs, which is critical for robust design optimisation and reliability analysis. However, standard interval propagation relies on solving optimisation problems that become computationally expensive for complex systems. Surrogate models alleviate this cost but typically replace only the evaluator within the optimisation loop, still requiring many inference calls. To overcome this limitation, we reformulate interval propagation as an interval-valued regression problem that directly predicts output bounds. We present a comprehensive study of neural network-based surrogate models, including multilayer perceptrons (MLPs) and deep operator networks (DeepONet), for this task. Three approaches are investigated: (i) naive interval propagation through standard architectures, (ii) bound propagation methods such as Interval Bound Propagation (IBP) and CROWN, and (iii) interval neural networks (INNs) with interval weights. Results show that these methods significantly improve computational efficiency over traditional optimisation-based approaches while maintaining accurate interval estimates. We further discuss practical limitations and open challenges in applying interval-based propagation methods.

Ghifari Adam Faza, Jolan Wauters, Fabio Cuzzolin +2

LMSD, Department of Mechanical Engineering, KU Leuven, Heverlee, 3001, Belgium · RAM, Department of Mechanical Engineering, KU Leuven, Brugge, 8200, Belgium · Visual Artificial Intelligence Laboratory, Oxford Brookes University, Oxford, UK +1

Uncertainty PropagationUncertainty QuantificationNeural Surrogate ModelingDeep Operator Networks
AskPDF
Mar 18, 2026

A Hybrid Conditional Diffusion-DeepONet Framework for High-Fidelity Stress Prediction in Hyperelastic Materials

Predicting stress fields in hyperelastic materials with complex microstructures remains challenging for traditional deep learning surrogates, which struggle to capture both sharp stress concentrations and the wide dynamic range of stress magnitudes. Convolutional architectures such as UNet tend to oversmooth high-frequency gradients, while neural operators like DeepONet exhibit spectral bias and underpredict localized extremes. Diffusion models can recover fine-scale structure but often introduce low-frequency amplitude drift, degrading physical scaling. To address these limitations, we propose a hybrid surrogate framework, cDDPM-DeepONet, that decouples stress morphology from magnitude. A conditional denoising diffusion probabilistic model (cDDPM), built on a UNet backbone, generates normalized von Mises stress fields conditioned on geometry and loading. In parallel, a modified DeepONet predicts global scaling parameters (minimum and maximum stress), enabling reconstruction of full-resolution physical stress maps. This separation allows the diffusion model to focus on spatial structure while the operator network corrects global amplitude, mitigating spectral and scaling biases. We evaluate the framework on nonlinear hyperelastic datasets with single and multiple polygonal voids. The proposed model consistently outperforms UNet, DeepONet, and standalone cDDPM baselines by one to two orders of magnitude. Spectral analysis shows strong agreement with finite element solutions across all wavenumbers, preserving both global behavior and localized stress concentrations.

Purna Vindhya Kota, Meer Mehran Rashid, Somdatta Goswami +1

Department of Civil and Systems Engineering, Johns Hopkins University, 3400 N. Charles Street, Baltimore, 21218, MD, USA

Neural Surrogate ModelingHyperelasticityDeep Operator NetworksConditional Diffusion Models
AskPDF
Mar 18, 2026

Translation Invariance of Neural Operators for the FitzHugh-Nagumo Model

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

Luca Pellegrini

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

Fourier Neural OperatorPDE Operator LearningDeep Operator NetworksNeural Operators
AskPDF
Mar 12, 2026

Topological DeepONets and a generalization of the Chen-Chen operator approximation theorem

Deep Operator Networks (DeepONets) provide a branch-trunk neural architecture for approximating nonlinear operators acting between function spaces. In the classical operator approximation framework, the input is a function u∈C(K1)u\in C(K_1) defined on a compact set K1K_1 (typically a compact subset of a Banach space), and the operator maps uu to an output function G(u)∈C(K2)G(u)\in C(K_2) defined on a compact Euclidean domain K2⊂RdK_2\subset\mathbb{R}^d. In this paper, we develop a topological extension in which the operator input lies in an arbitrary Hausdorff locally convex space XX. We construct topological feedforward neural networks on XX using continuous linear functionals from the dual space X∗X^* and introduce topological DeepONets whose branch component acts on XX through such linear measurements, while the trunk component acts on the Euclidean output domain. Our main theorem shows that continuous operators G:V→C(K;Rm)G:V\to C(K;\mathbb{R}^m), where V⊂XV\subset X and K⊂RdK\subset\mathbb{R}^d are compact, can be uniformly approximated by such topological DeepONets. This extends the classical Chen-Chen operator approximation theorem from spaces of continuous functions to locally convex spaces and yields a branch-trunk approximation theorem beyond the Banach-space setting.

Vugar Ismailov

Neural Network Approximation TheoryDeep Operator NetworksNeural Operators
AskPDF
Mar 9, 2026

Tau-BNO: Brain Neural Operator for Tau Transport Model

Mechanistic modeling provides a biophysically grounded framework for studying the spread of pathological tau protein in tauopathies like Alzheimer's disease. Existing approaches typically model tau propagation as a diffusive process on the brain's structural connectome, reproducing macroscopic patterns but neglecting microscale cellular transport and reaction mechanisms. The Network Transport Model (NTM) was introduced to fill this gap, explaining how region-level progression of tau emerges from microscale biophysical processes. However, the NTM faces a common challenge for complex models defined by large systems of partial differential equations: the inability to perform parameter inference and mechanistic discovery due to high computational burden and slow model simulations. To overcome this barrier, we propose Tau-BNO, a Brain Neural Operator surrogate framework for rapidly approximating NTM dynamics that captures both intra-regional reaction kinetics and inter-regional network transport. Tau-BNO combines a function operator that encodes kinetic parameters with a query operator that preserves initial state information, while approximating anisotropic transport through a spectral kernel that retains directionality. Empirical evaluations demonstrate high predictive accuracy (R2≈R^2\approx 0.98) across diverse biophysical regimes and an 89% performance improvement over state-of-the-art sequence models like Transformers and Mamba, which lack inherent structural priors. By reducing simulation time from hours to seconds, we show that the surrogate model is capable of producing new insights and generating new hypotheses. This framework is readily extensible to a broader class of connectome-based biophysical models, showcasing the transformative value of deep learning surrogates to accelerate analysis of large-scale, computationally intensive dynamical systems.

Nuutti Barron, Heng Rao, Urmi Saha +6

Department of Radiology and Biomedical Imaging, University of California San Francisco, USA · College of Computer Science and Engineering, Northeastern University, China · Department of Computer Science, Wake Forest University, USA +1

Neural Surrogate ModelingPDE Surrogate ModelingNeural OperatorsDeep Operator Networks
AskPDF
Mar 2, 2026

Tackling multiphysics problems via finite element-guided physics-informed operator learning

This work presents a finite element-guided physics-informed operator learning framework for multiphysics problems with coupled partial differential equations (PDEs) on arbitrary domains. The proposed framework learns an operator from the input space to the solution space with a weighted residual formulation based on the finite element method, enabling discretization-independent prediction beyond the training resolution without relying on labeled simulation data. The present framework for multiphysics problems is implemented in Folax, a JAX-based operator learning platform, and is verified on nonlinear coupled thermo-mechanical problems. Two- and three-dimensional representative volume elements with varying heterogeneous microstructures, and a close-to-reality industrial casting example under varying boundary conditions are investigated as the example problems. We investigate the potential of several neural operators combined with the proposed finite element-guided approach, including Fourier neural operators (FNOs), deep operator networks (DeepONets), and a newly proposed implicit finite operator learning (iFOL) approach based on conditional neural fields. The results demonstrate that FNOs yield highly accurate solution operators on regular domains, where the global features can be efficiently learned in the spectral domain, and iFOL offers efficient parametric operator learning capabilities for complex and irregular geometries. Furthermore, studies on training strategies, network decomposition, and training sample quality reveal that a monolithic training strategy using a single network is sufficient for accurate predictions, while training sample quality strongly influences performance. Overall, the present approach highlights the potential of physics-informed operator learning with a finite element-based loss as a unified and scalable approach for coupled multiphysics simulations.

Yusuke Yamazaki, Reza Najian Asl, Markus Apel +2

Graduate School of Science and Technology, Keio University, Hiyoshi3-14-1,Kohoku-ku, Yokohama 223-8522, Japan · Chair of Structural Analysis, Technical University of Munich, Arcisstraße 21, 80333 Munich, Germany · ACCESS e.V., Intzestr. 5, 52072 Aachen, Germany +1

Multiphysics SimulationPhysics-Informed Neural OperatorsPhysics-Based SimulationDeep Operator NetworksFourier Neural OperatorImplicit Neural Representations
AskPDF
Feb 28, 2026

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

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

Hongquan Wang, Hanshu Chen, Ilia Marchevsky +1

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

PDE Operator LearningDeep Operator NetworksHybrid Quantum-Classical ML
AskPDF
Feb 25, 2026

The Error of Deep Operator Networks Is the Sum of Its Parts: Branch-Trunk and Mode Error Decompositions

Operator learning has the potential to strongly impact scientific computing by learning solution operators for differential equations, potentially accelerating multi-query tasks such as design optimization and uncertainty quantification by orders of magnitude. Despite proven universal approximation properties, deep operator networks (DeepONets) often exhibit limited accuracy and generalization in practice, which hinders their adoption. Understanding these limitations is therefore crucial for further advancing the approach. This work analyzes performance limitations of the classical DeepONet architecture. It is shown that the approximation error is dominated by the branch network when the internal dimension is sufficiently large, and that the learned trunk basis can often be replaced by classical basis functions without a significant impact on performance. To investigate this further, a modified DeepONet is constructed in which the trunk network is replaced by the left singular vectors of the training solution matrix. This modification yields several key insights. First, for examples involving the KdV and Burgers equations, a spectral bias in the branch network is observed, with coefficients of dominant, low-frequency modes learned more effectively. Second, through the interplay of the singular-value weighting and the optimizer's neglect of small modes, the branch error is dominated by modes with large and intermediate singular values. Third, using a shared branch network for all mode coefficients, as in the standard architecture, improves generalization of small modes compared to a stacked architecture in which coefficients are computed separately. Finally, detrimental coupling between modes in parameter space is identified, which weakens with increasing network width.

Alexander Heinlein, Johannes Taraz

Delft Institute of Applied Mathematics, Delft University of Technology, Delft, 2628CD, The Netherlands

Deep Operator NetworksNeural OperatorsNeural Network GeneralizationNeural Network Training Dynamics
AskPDF
Dec 9, 2025

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 L2L^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 ModelingPDE Operator LearningSpectral MethodsDeep Operator Networks
AskPDF
Jul 29, 2025

PVD-ONet: A Multi-scale Neural Operator Method for Singularly Perturbed Boundary Layer Problems

Physics-informed neural networks and physics-informed DeepONet excel at solving partial differential equations but often struggle with singularly perturbed problems. To address this limitation, we propose two novel data-free frameworks: the Prandtl--Van Dyke neural network(PVD-Net) and its operator-learning extension, the Prandtl--Van Dyke Deep Operator Network(PVD-ONet). Each framework provides leading-order and high-order versions to meet different modeling requirements. The leading-order PVD-Net integrates Prandtl's matching condition for improved stability, while the high-order PVD-Net employs Van Dyke's matching principle to capture fine-scale boundary layer structures with higher accuracy. Extending PVD-Net to operator learning, PVD-ONet enables efficient prediction for families of boundary layer problems without retraining. Numerical experiments on representative one- and two-dimensional problems demonstrate that the proposed methods outperform existing baselines. Beyond forward prediction, the framework also supports inverse problems by inferring the scaling exponent governing boundary layer thickness from sparse observations, highlighting its potential for practical applications.

Tiantian Sun, Jian Zu

School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, P.R. China

Neural OperatorsDeep Operator Networks
AskPDF
Mar 31, 2025

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 pp-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 TheoryDeep Operator NetworksOperator Learning
AskPDF
Mar 7, 2025

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 ModelingFourier Neural OperatorPDE Operator LearningDeep Operator NetworksNeural Operators
AskPDF
Dec 15, 2021

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 dd-dimensional periodic domains, d=1,2,…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 H1H^1 norm, uniformly over the coefficient set. We prove that the neural networks in the ONet have size O(∣log⁡(ε)∣κ)\mathcal{O}(\left|\log(\varepsilon)\right|^κ), where ε>0\varepsilon>0 is the approximation accuracy, for some κ>0κ>0 depending on the physical space dimension.

Carlo Marcati, Christoph Schwab

Seminar for Applied Mathematics, ETH Zurich, Switzerland

PDE SolvingPDE Operator LearningNeural Network Approximation TheoryDeep Operator Networks
AskPDF