TL;DR
DeepONet learns nonlinear operators by combining a branch net (encoding the input function) and a trunk net (encoding the query location), enabling supervised operator learning with a universal approximation guarantee.
Problem
Approximate solution operators of differential equations: map an input function (IC/BC/forcing/coefficient) to a solution function, so that inference becomes a fast forward pass instead of running a numerical solver.
Benefits vs others
- General operator viewpoint: one model can answer many queries (many right-hand-sides / coefficients / boundary conditions) once trained.
- Efficient inference: after training, evaluation at new coordinates is cheap (trunk net) and re-usable for many y points.
- Theory-backed: universal approximation result for continuous operators motivates architecture.
Interesting detail
- Because the trunk net depends only on y, it can be precomputed for a fixed output grid; then many inputs u can be evaluated cheaply by changing only the branch output.
- The dot-product form can be interpreted as learning a data-driven basis expansion for the solution operator.
Core method (math)
Template for Operator learning. Paper-specific equations are added when manually curated.
\[G_\theta(u)(y) = \sum_{k=1}^{p} b_k(u)\, t_k(y)\quad\text{(branch outputs }b_k,\ \text{trunk outputs }t_k)\]
\[\hat s(y) = \langle \mathbf{b}(u),\ \mathbf{t}(y) \rangle\ (+\ b_0)\quad\text{(dot-product readout)}\]
Main theoretical contribution
- Universal approximation (operator version): for suitable compact sets of input functions and continuous operators G, there exist network parameters such that DeepONet approximates G uniformly to arbitrary accuracy.
Main contribution
- Propose the Deep Operator Network (DeepONet) architecture: a branch network consumes samples of an input function u and outputs coefficients; a trunk network consumes query coordinates y and outputs basis values; the output is a dot product.
- Provide a universal approximation theorem for operators (under continuity assumptions), motivating DeepONet as a principled operator learner.
- Demonstrate DeepONet on challenging operator-learning tasks including implicit operators in a diffusion–reaction system and stochastic PDEs, achieving orders-of-magnitude lower test error than strong neural baselines in reported setups.
Main results (headline)
(Optional) Add main_results for a quick headline summary.
Experiments
PDE problems
- Diffusion–reaction system
- Stochastic PDEs
Tasks
- Operator learning / surrogate modeling
- Fast PDE surrogate inference
Experiment setting (high level)
- Learns mapping from an input function (IC/forcing) to output function values.
- Often evaluated with scattered sensor inputs and queried outputs.
Comparable baselines
- FNN (fully-connected neural network baseline in paper)
- ResNet (baseline in paper)
Main results
Reported quantitative highlights (from the paper text; see Fig. 4 and Supplementary Sec. 6)
These numbers are quoted from the paper’s reported best results (not an exhaustive benchmark).
| Task | Metric | DeepONet | Baseline(s) |
|---|---|---|---|
| Diffusion–reaction system (implicit operator) | Smallest test error | ≈ 1e-5 | ResNet ≈ 1e-1 (same setting) |
| Stochastic PDE (multiplicative noise) | Best test MSE / rel. L2 | MSE=1e-5; L2=1.1% | — |
| Stochastic PDE (additive noise) | Best test MSE / rel. L2 | MSE=9.4e-4; L2=0.03% | — |
Citation (BibTeX)
@article{lu2021deeponet,
title={Learning nonlinear operators via {DeepONet} based on the universal approximation theorem of operators},
author={Lu, Lu and Jin, Pengzhan and Karniadakis, George Em},
journal={Nature Machine Intelligence},
year={2021}
}