Proves the first universal approximation theorems for k-times differentiable nonlinear operators between Banach spaces and their derivatives uniformly on compact sets in weighted Sobolev norms via encoder-decoder operator learning architectures.
hub Canonical reference
DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
Canonical reference. 77% of citing Pith papers cite this work as background.
40
Pith papers citing it
Background
77% of classified citations
abstract
While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensors $x_i, i=1,\dots,m$ (branch net), and another for encoding the locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset size.
hub tools
citation-role summary
background 11
method 2
citation-polarity summary
representative citing papers
Constraint-Aware Flow Matching integrates constraint projections into the flow matching training objective to align model dynamics with constrained sampling and reduce distributional shift.
Any maximally monotone operator can be approximated in local graph convergence by continuous encoder-decoder networks, with structure-preserving versions that retain maximal monotonicity via resolvent parameterizations.
A latent Structured Spectral Propagator enables stable autoregressive PDE forecasting by decoupling spatial details from recurrent modal dynamics.
CATO learns a continuous latent chart for efficient axial attention on PDE meshes and adds derivative-aware supervision to improve accuracy and reduce oversmoothing on general geometries.
Spatio-Temporal MeanFlow adapts MeanFlow to PDEs by replacing the generative velocity field with the physical operator and extending the integral constraint to the spatio-temporal domain, yielding a unified solver for time-dependent and stationary equations with improved accuracy and generalization.
GANO is an end-to-end differentiable latent-space optimizer that unifies shape encoding, surrogate prediction, and controllable geometry updates for PDE-governed shape optimization and inversion.
AI models of viscous fingering exhibit hallucinations from spectral bias; DeepFingers combines FNO and DeepONet with time-contrast conditioning to predict accurate finger dynamics while preserving mixing metrics.
A DeepRitzSplit neural operator trained on energy-split variational forms enforces dissipation in phase-field models and outperforms data-driven training in generalization while running faster than Fourier spectral methods on Allen-Cahn and dendritic growth cases.
DiLO turns diffusion sampling into deterministic latent optimization to satisfy the manifold consistency requirement for neural operators in inverse problem solving.
Flow matching on time series targets a closed-form nonparametric velocity field that is a similarity-weighted mixture of observed transition velocities, making neural models approximations to an ideal memory-augmented dynamical system sampler.
CompNO composes specialized Fourier neural operator blocks for fundamental differential operators into task-specific solvers that achieve lower L2 error than baselines on linear parametric PDEs and remain competitive on nonlinear flows while exactly satisfying boundaries.
Transformers and generalized neural integral operators are shown to universally approximate operators between Hölder and Banach spaces.
Symplectic Neural Operators preserve symplectic structure for learning infinite-dimensional Hamiltonian PDEs and deliver improved long-term energy stability in theory and experiments.
Compositional Neural Operators decompose multi-dimensional fluid PDEs into a library of pretrained elementary physics blocks assembled via an aggregator that minimizes data and physics residuals.
ABLE learns a spatially adaptive Parseval frame from data via an ancillary density to replace fixed bases in spectral neural operators for PDEs.
S4 models exhibit stable time-continuity unlike sensitive S6 models, with task continuity predicting performance and enabling temporal subsampling for better efficiency.
MuFiNNs integrates sparse experimental measurements with structured low-fidelity models via hierarchical construction and nonlinear correction to predict 3D flame wrinkling dynamics and turbulent mass burning velocity across fuels, pressures, and turbulence levels.
Late Fusion Neural Operators disentangle state and parameter learning to outperform FNO and CAPE-FNO on advection, Burgers, and reaction-diffusion PDEs with 72% average RMSE reduction in and out of domain.
Hypernetworks map a forcing parameter directly to policy weights in an RL framework, enabling unified stabilization of the Kuramoto-Sivashinsky equation across regimes with KAN architectures showing strongest extrapolation.
A certified adaptive quadrature framework computes guaranteed L^p, W^{1,p}, and W^{2,p} norms of deep neural networks by propagating interval enclosures on axis-aligned boxes.
GSNO uses position-dependent spherical Green's functions to create flexible neural operators that adapt to non-equivariant systems on spheres while keeping spectral efficiency and grid invariance.
DIANO builds coarse-grid latent spaces for fluid dynamics data via neural operator encoding and decoding while integrating a differentiable PDE solver directly in the latent space for end-to-end physics-constrained training.
Neural networks regress oversized subspaces for parametric problems using subspace-specific losses, with theory and experiments showing improved accuracy and smoother mappings.
citing papers explorer
-
Generalized Spherical Neural Operators: Green's Function Formulation
GSNO uses position-dependent spherical Green's functions to create flexible neural operators that adapt to non-equivariant systems on spheres while keeping spectral efficiency and grid invariance.
-
Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling
DIANO builds coarse-grid latent spaces for fluid dynamics data via neural operator encoding and decoding while integrating a differentiable PDE solver directly in the latent space for end-to-end physics-constrained training.
-
Deep Learning for Subspace Regression
Neural networks regress oversized subspaces for parametric problems using subspace-specific losses, with theory and experiments showing improved accuracy and smoother mappings.
-
Latent Space Dynamics Identification for Interface Tracking with Application to Shock-Induced Pore Collapse
LaSDI-IT learns latent linear dynamics for interface tracking via a revised autoencoder and Gaussian process interpolation, achieving under 9% error and 106x speedup on shock-induced pore collapse in high explosives.
-
Operator Learning for Schr\"{o}dinger Equation: Unitarity, Error Bounds, and Time Generalization
A linear estimator for the Schrödinger evolution operator is introduced that enforces weak unitarity, supplies uniform prediction error bounds and time-extrapolation bounds, and reports up to 100x lower relative error than FNO and DeepONet on hydrogen, ion-trap, and optical-lattice Hamiltonians.
-
On the definition and importance of interpretability in scientific machine learning
Interpretability in SciML requires mechanistic understanding rather than sparsity, and prior knowledge is often essential for interpretable scientific discovery.
-
Teaching Artificial Intelligence to Perform Rapid, Resolution-Invariant Grain Growth Modeling via Fourier Neural Operator
FNO surrogate model learns to predict long-term grain growth evolution from phase-field data while remaining accurate on unseen configurations and higher-resolution grids.
-
ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithms
ATHENA introduces an agentic team framework that autonomously manages the end-to-end computational research lifecycle via a knowledge-driven HENA loop to achieve validation errors of 10^{-14} in scientific computing and machine learning tasks.
-
XRePIT: A deep learning-computational fluid dynamics hybrid framework implemented in OpenFOAM for fast, robust, and scalable unsteady simulations
XRePIT automates residual-guided switching between neural surrogates and OpenFOAM to enable stable, up to 2.91x faster 3D unsteady flow simulations with L2 errors around 1E-03.
-
A Practitioner's Guide to Kolmogorov-Arnold Networks
A systematic review of Kolmogorov-Arnold Networks that maps their relation to Kolmogorov superposition theory, MLPs, and kernels, examines basis-function design choices, summarizes performance advances, and supplies a practitioner's selection guide plus open challenges.