Deep Neural ODE Operator Networks for PDEs
Ziqian Li, Kang Liu, Yongcun Song, Hangrui Yue, Enrique Zuazua
TL;DR
The paper addresses fast, accurate operator learning for PDEs by embedding domain knowledge into a neural ODE framework. It introduces NODE-ONet, an encoder–decoder architecture where a spatial encoder reduces inputs to a latent state, a physics-encoded neural ODE evolves this latent state, and a decoder reconstructs the solution $u(t,x)$ in physical space. A general encoder–decoder error-analysis framework is developed, and physics-encoded NODEs are designed to respect PDE structure (e.g., bilinear diffusion and additive forcing) to improve efficiency and generalization, including extrapolation beyond training time. Numerical experiments on nonlinear diffusion–reaction and 2D Navier–Stokes demonstrate high accuracy, computational efficiency, and the ability to generalize across related PDEs, with comparisons showing NODE-ONet outperforming traditional DeepONet and MIONet baselines in multi-input settings. The framework is flexible, enabling various encoders/decoders and potential extensions to stationary cases and other PDE families, offering a scalable, physics-informed surrogate tool for scientific machine learning.
Abstract
Operator learning has emerged as a promising paradigm for developing efficient surrogate models to solve partial differential equations (PDEs). However, existing approaches often overlook the domain knowledge inherent in the underlying PDEs and hence suffer from challenges in capturing temporal dynamics and generalization issues beyond training time frames. This paper introduces a deep neural ordinary differential equation (ODE) operator network framework, termed NODE-ONet, to alleviate these limitations. The framework adopts an encoder-decoder architecture comprising three core components: an encoder that spatially discretizes input functions, a neural ODE capturing latent temporal dynamics, and a decoder reconstructing solutions in physical spaces. Theoretically, error analysis for the encoder-decoder architecture is investigated. Computationally, we propose novel physics-encoded neural ODEs to incorporate PDE-specific physical properties. Such well-designed neural ODEs significantly reduce the framework's complexity while enhancing numerical efficiency, robustness, applicability, and generalization capacity. Numerical experiments on nonlinear diffusion-reaction and Navier-Stokes equations demonstrate high accuracy, computational efficiency, and prediction capabilities beyond training time frames. Additionally, the framework's flexibility to accommodate diverse encoders/decoders and its ability to generalize across related PDE families further underscore its potential as a scalable, physics-encoded tool for scientific machine learning.