Time-Invariant Neural Operators with Applications in Solving Time-Dependent PDEs

Jul 16, 2026

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DeepONet is a foundational architecture for learning nonlinear operators with neural networks, yet it does not naturally enforce key temporal properties for dynamic physical responses. This work introduces time-invariant neural operator (TINO) by linking time-delay neural network (TDNN) with DeepONet to respect time invariance and address limitations related to time causality. The framework is extended to initial-value time-dependent PDEs via truncated time-invariant formulations with spatial POD in the output function space. Numerical experiments validate improved accuracy over causality-aware and causality-free operator learning baselines.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages30

Reading Time1 h 16 min

Introduction

Time causality and time invariance

Limitations of operator learning for temporal properties

Related neural operator approaches

Paper contributions and organization

TINO and truncated formulations

arXiv :2607 .06188v1 [math .NA] 7 Jul 2026 Time-Invariant Neural Operators with Applications in Solving Time-Dependent PDEs Zihan Zhoua,b,1 , Wenzhong Zhangb,c,1 , Lizuo Liud,∗ a University of Science and Technology of China, 230027, Hefei, P.R. China b Suzhou Institute for Advanced Research, University of Science and Technology of China, 215123, Suzhou, P. R. China c Key Laboratory of the Ministry of Education for Mathematical Foundations and Applications of Digital Technology, 230027, Hefei, P.R. China dDartmouth College, 03755, Hanover, NH, USA Abstract The deep operator network (DeepONet) is one of the basic architectures for learning nonlinear operators with neural networks. However, for operators that describe the dynamic response of physical systems, DeepONet does not naturally respect fundamental time properties, including time causality and time invariance. We propose time-invariant neural operator (TINO) to overcome these limitations, by creating the connection between time-delay neural network (TDNN) and DeepONet. We then extend the proposal to solving timedependent PDEs with initial values, which can be treated as truncated time-invariant systems, with spatial proper orthogonal decomposition (POD) implementation in the output function space. Various numerical tests comparing TINO, neural operators with time causality, and those without time properties justify the enhanced precision of the proposed neural operator frameworks. 1. Introduction Learning the dynamic response of physical systems is a fundamental task in engineering and in scientific computing, with broad applications in control theory, signal processing, seismology, etc. In many applications, the object of interest is the operator that maps a time-varying input signal, forcing term, or source field to a response function over time and space. Two foundational temporal properties, time causality and time invariance, play fundamental roles to understanding and modeling these systems. Time causality describes that the output at any time t depends only on present and past input, whereas time invariance requires that a temporal shift of the input produces the corresponding shift of the output. In signal processing, time invariance is equivalently described as shift invariance, which indicates the commutative property that ensures consistent behavior regardless of arbitrary shifting operations. These temporal properties are well understood in classic frameworks. Linear time-invariant (LTI) systems admit convolution representations, while nonlinear operators are related to Volterra series, Wiener-type expansions, and state-space formulations, to name a few. In terms of PDE solvers for finding single trajectories, numerical solvers are typically built in a time marching manner to incrementally solve the output by ∗ Corresponding author. Email addresses: [Lizuo.Liu@Dartmouth.edu](Lizuo.Liu@Dartmouth.edu) (LL), [zihanzhou@mail.ustc.edu.cn](zihanzhou@mail.ustc.edu.cn) (ZZ), [wenzhong@ustc.edu.cn](wenzhong@ustc.edu.cn) (WZ). 1 Equal contribution. ascending time steps, hence strictly obeying time causality. However, these temporal properties are not automatically inherited by data-driven operator learning methods. Since the models are learned directly from data, it raises a nontrivial modeling problem to preserve temporal properties for prediction tasks. Namely, time causality rejects future input to the model, and time invariance prohibits the model from explicitly depending on time. In recent years, advancements in deep learning have made operator learning powerful and versatile for modeling dynamical systems. Approaches such as recurrent neural networks (RNN) and their variants, Mori–Zwanzig Net [6], OnsargarNet [37] and the Laplace neural operator with a convolution kernel [1], Causality-DeepONet [14], Causal-DeepONet [18], Mamba and other state-space models [7, 39, 9, 4, 5, 8], etc. have demonstrated significant potential in reinte

FAQs

What limitation of DeepONet motivates the proposed TINO framework?

DeepONet does not naturally respect fundamental time properties, particularly time causality and time invariance, which are essential for operators describing dynamic physical responses.

How does TINO enforce time invariance in neural operator learning?

TINO constructs a time-discretized, time-invariant neural operator by integrating DeepONet with a time-delay embedding, encoding temporal dependencies through delayed inputs while keeping the output basis independent of absolute time.

How is TINO extended to solve initial-value problems for time-dependent PDEs?

The method interprets the initial-value setting as a truncated evolution of an underlying time-invariant system, then applies a time-truncated TINO formulation with spatial proper orthogonal decomposition (POD) in the output function space.

Jul 16, 2026

Upload by Miles