RHYME-XT: A Neural Operator for Spatiotemporal Control Systems

TL;DR

RHYME-XT introduces a neural operator framework for efficient surrogate modeling of complex spatiotemporal control systems governed by nonlinear PIDEs, enabling continuous-time, discretization-invariant predictions.

Contribution

It presents a novel neural operator that combines Galerkin projection with flow map learning to model nonlinear PIDEs without costly integrations.

Findings

Outperforms existing neural operators on neural field PIDEs.

Effectively transfers knowledge across models via fine-tuning.

Provides continuous-time, discretization-invariant solutions.

Abstract

We propose RHYME-XT, an operator-learning framework for surrogate modeling of spatiotemporal control systems governed by input-affine nonlinear partial integro-differential equations (PIDEs) with localized rhythmic behavior. RHYME-XT uses a Galerkin projection to approximate the infinite-dimensional PIDE on a learned finite-dimensional subspace with spatial basis functions parameterized by a neural network. This yields a projected system of ODEs driven by projected inputs. Instead of integrating this non-autonomous system, we directly learn its flow map using an architecture for learning flow functions, avoiding costly computations while obtaining a continuous-time and discretization-invariant representation. Experiments on a neural field PIDE show that RHYME-XT outperforms a state-of-the-art neural operator and is able to transfer knowledge effectively across models trained on different datasets, through a fine-tuning process.