[Paper Review] Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets

Sifan Wang, Hanwen Wang+1 more|arXiv (Cornell University)|Mar 19, 2021

Model Reduction and Neural Networks61 citations

TL;DR

The paper extends DeepONet with physics-informed regularization to learn parametric PDE solution operators, enabling data-efficient training and PDE-consistent predictions, including zero-shot data-free learning given only initial/boundary conditions, and accelerates PDE solving by orders of magnitude.

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}.

Motivation & Objective

• Motivate learning solution operators for parametric PDEs and address data efficiency and physical consistency shortcomings of standard DeepONets.
• Introduce a physics-informed regularization framework that enforces governing PDEs via automatic differentiation during training.
• Demonstrate data efficiency and zero-shot learning capabilities for operator learning under limited or no paired data.
• Showcase performance across diffusion-reaction, ODEs, and high-frequency input regimes, including out-of-distribution tests.

Proposed method

• Represent the solution map G as an unstacked DeepONet with a branch net for inputs and a trunk net for coordinates, merged by a dot product.
• Train with a composite loss L = L_operator + L_physics, where L_operator matches DeepONet outputs to true solutions (when available).
• Formulate L_physics as a PDE residual penalty using automatic differentiation to enforce governing laws.
• Use Gaussian random field inputs to generate diverse parametric PDE scenarios and assess data efficiency and generalization.
• In challenging cases, employ Fourier feature embeddings to capture high-frequency content in inputs and outputs.
• Demonstrate data-free learning where the model respects initial/boundary conditions and PDE constraints without paired outputs.

Experimental results

Research questions

• RQ1 Can physics-informed regularization guide DeepONets to respect underlying PDEs without large paired datasets?
• RQ2 How does physics-informed training affect data efficiency, generalization, and extrapolation for parametric PDE operators?
• RQ3 What architectures (e.g., Fourier features) enable learning high-frequency solution components from irregular inputs?
• RQ4 To what extent can such models accelerate PDE solving compared to conventional solvers?
• RQ5 Are zero-shot or data-light trainings feasible for complex parametric PDEs with realistic IBCs?

Key findings

• Physics-informed DeepONets significantly improve predictive accuracy and reduce data requirements compared to conventional DeepONets.
• The physics-informed framework enables learning operators without paired input-output data, relying on initial/boundary conditions and PDE residuals.
• With appropriate architectures (e.g., Fourier features), the method accurately handles high-frequency input functions and irregular inputs.
• Predicted solutions and residuals align well with ground truth, and zero-shot or data-sparse training achieves competitive accuracy.
• Physics-informed DeepONets can predict PDE solutions orders of magnitude faster than conventional solvers.