Learning PDE Solution Operator for Continuous Modeling of Time-Series

TL;DR

This paper introduces a PDE-based neural operator framework that models time-continuous dynamics efficiently, with theoretical universality and improved stability, outperforming existing models in real-time series applications.

Contribution

It presents a novel neural operator capable of continuous time modeling without iterative steps, with proven universality and enhanced stability for better generalization.

Findings

Achieves superior accuracy on time-dependent PDEs

Demonstrates better data efficiency and generalization

Outperforms state-of-the-art models in real-time series tasks

Abstract

Learning underlying dynamics from data is important and challenging in many real-world scenarios. Incorporating differential equations (DEs) to design continuous networks has drawn much attention recently, however, most prior works make specific assumptions on the type of DEs, making the model specialized for particular problems. This work presents a partial differential equation (PDE) based framework which improves the dynamics modeling capability. Building upon the recent Fourier neural operator, we propose a neural operator that can handle time continuously without requiring iterative operations or specific grids of temporal discretization. A theoretical result demonstrating its universality is provided. We also uncover an intrinsic property of neural operators that improves data efficiency and model generalization by ensuring stability. Our model achieves superior accuracy in dealing with time-dependent PDEs compared to existing models. Furthermore, several numerical pieces of evidence validate that our method better represents a wide range of dynamics and outperforms state-of-the-art DE-based models in real-time-series applications. Our framework opens up a new way for a continuous representation of neural networks that can be readily adopted for real-world applications.