Deep Neural ODE Operator Networks for PDEs

2025/10/17 by Ziqian Li, Kang Liu, Li, Ziqian +7
Materials Science · Mathematics · Physics and Astronomy · #47-08 #65D15 #68T07 #AMS Subject Classification: 65M99 #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning in Materials Science #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2510.15651

openalex publication_date 2025/10/17 · openalex created_date 2025/10/21 · openalex updated_date 2026/07/28

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.

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