Time Resolution Independent Operator Learning

TL;DR

This paper introduces NCDE-DeepONet, a continuous-time operator learning framework that encodes entire load histories as input-independent representations, enabling accurate, resolution-independent predictions for time-dependent PDEs from sparse data.

Contribution

The paper proposes NCDE-DeepONet, integrating Neural Controlled Differential Equations into operator learning to achieve input and output resolution independence for transient PDEs.

Findings

Achieves accurate predictions on transient Poisson, elastodynamic, and thermoelastic problems.

Enables instant solution predictions without retraining or interpolation.

Robustness confirmed across multiple benchmark problems.

Abstract

Accurately learning solution operators for time-dependent partial differential equations (PDEs) from sparse and irregular data remains a challenging task. Recurrent DeepONet extensions inherit the discrete-time limitations of sequence-to-sequence (seq2seq) RNN architectures, while neural-ODE surrogates cannot incorporate new inputs after initialization. We introduce NCDE-DeepONet, a continuous-time operator network that embeds a Neural Controlled Differential Equation (NCDE) in the branch and augments the trunk with explicit space-time coordinates. The NCDE encodes an entire load history as the solution of a controlled ODE driven by a spline-interpolated input path, making the representation input-resolution-independent: it encodes different input signal discretizations of the observed samples. The trunk then probes this latent path at arbitrary spatial locations and times, rendering the overall map output-resolution independent: predictions can be queried on meshes and time steps unseen during training without retraining or interpolation. Benchmarks on transient Poisson, elastodynamic, and thermoelastic problems confirm the robustness and accuracy of the framework, achieving almost instant solution prediction. These findings suggest that controlled dynamics provide a principled and efficient foundation for high-fidelity operator learning in transient mechanics.