RHYME-XT: A Neural Operator for Spatiotemporal Control Systems

2026-03-18

AI Summary

Existing methods struggle to efficiently and discretization-independently model spatiotemporal systems governed by input-affine nonlinear partial integro-differential equations (PIDEs) with local rhythmic behavior. This work proposes the RHYME-XT framework, which uniquely integrates Galerkin projection with continuous-time flow map learning. Specifically, neural network–parameterized spatial basis functions are employed to construct a finite-dimensional subspace; projecting the PIDE onto this subspace yields a system of ordinary differential equations whose continuous-time flow map is learned directly, bypassing explicit numerical integration. This approach enables efficient, discretization-independent modeling and facilitates knowledge transfer across datasets. Evaluated on neural field PIDE tasks, RHYME-XT substantially outperforms existing neural operator methods.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• spatiotemporal control systems
• nonlinear partial integro-differential equations
• surrogate modeling
• localized rhythmic behavior
• input-affine PIDEs

Innovation

• neural operator
• Galerkin projection
• flow map learning
• spatiotemporal control
• PIDE