Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets

TL;DR

This work tackles learning the solution operator of parametric PDEs between infinite‑dimensional spaces by extending DeepONets with physics‑informed regularization. The core idea is to train the operator network $G_{\bm{\theta}}(\bm{u})(\bm{y})$ not only to match observed input–output pairs but also to satisfy the governing PDE via residuals computed by automatic differentiation, yielding a total loss $\mathcal{L}(\bm{\theta}) = \mathcal{L}_{\text{operator}}(\bm{\theta}) + \mathcal{L}_{\text{physics}}(\bm{\theta})$. The approach achieves up to 1–2 orders of magnitude improvement in predictive accuracy, enables solving parametric PDEs with no paired data (except IBCs), and delivers speedups of up to three orders of magnitude in inference compared to traditional solvers. Across a range of benchmarks—including anti‑derivative, diffusion–reaction, Burgers', and Eikonal equations—the method demonstrates strong data efficiency, generalization, and robustness to irregular inputs. These results suggest a broad impact for accelerating scientific computation and engineering design with physics‑consistent operator learning.

Abstract

Deep operator networks (DeepONets) are receiving increased attention thanks to their demonstrated capability to approximate nonlinear operators between infinite-dimensional Banach spaces. However, despite their remarkable early promise, they typically require large training data-sets consisting of paired input-output observations which may be expensive to obtain, while their predictions may not be consistent with the underlying physical principles that generated the observed data. In this work, we propose a novel model class coined as physics-informed DeepONets, which introduces an effective regularization mechanism for biasing the outputs of DeepOnet models towards ensuring physical consistency. This is accomplished by leveraging automatic differentiation to impose the underlying physical laws via soft penalty constraints during model training. We demonstrate that this simple, yet remarkably effective extension can not only yield a significant improvement in the predictive accuracy of DeepOnets, but also greatly reduce the need for large training data-sets. To this end, a remarkable observation is that physics-informed DeepONets are capable of solving parametric partial differential equations (PDEs) without any paired input-output observations, except for a set of given initial or boundary conditions. We illustrate the effectiveness of the proposed framework through a series of comprehensive numerical studies across various types of PDEs. Strikingly, a trained physics informed DeepOnet model can predict the solution of $\mathcal{O}(10^3)$ time-dependent PDEs in a fraction of a second -- up to three orders of magnitude faster compared a conventional PDE solver. The data and code accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/Physics-informed-DeepONets}.

Figures (28)

• Figure 1: Making DeepOnets physics-informed: The DeepONet architecture lu2019deeponet consists of two sub-networks, the branch net for extracting latent representations of input functions, and the trunk net for extracting latent representations of input coordinates at which the output functions are evaluated. A continuous and differentiable representation of the output functions is then obtained by merging the latent representations extracted by each sub-network via a dot product. Automatic differentiation can then be employed to formulate appropriate regularization mechanisms for biasing the DeepOnet outputs to satisfy a given system of PDEs.
• Figure 2: Learning the anti-derivative operator: Predicted solution $s(x)$ and residual $u(x)$ versus the ground truth for a representative input function. The results are obtained by training a conventional DeepONet model lu2019deeponet equipped with different activation functions after 40,000 iterations of gradient descent using the Adam optimizer.
• Figure 3: Learning anti-derivative operator: Exact solution and residual versus the predictions of a trained physics-informed DeepONet for the same input function as in Figure \ref{['fig: deeponet_antideriv_s_u']}.
• Figure 4: Learning anti-derivative operator: Mean of the relative $L^2$ prediction error of the original DeepONet lu2019deeponet and the physics-informed DeepONet as a function of the number of $u$ samples.
• Figure 5: Solving a 1D parametric ODE: (a)(b) Exact solution and residual versus the predictions of a trained physics-informed DeepONet for a representative input sample.

Theorems & Definitions (1)

• Remark 2.1