TRENDy: Temporal Regression of Effective Non-linear Dynamics

2024-12-04

arXiv.org

AI Summary

This work addresses the challenge of identifying and predicting bifurcation points (e.g., Turing, Hopf) in spatiotemporal dynamical systems governed by unknown partial differential equations (PDEs), under conditions of high noise and sparse sampling. Method: We propose an equation-free learning framework that jointly employs multi-scale filtering for dimensionality reduction and parameter-aligned neural ODEs to model effective dynamics on a low-dimensional manifold. Crucially, we introduce the first coupling of phase-space geometric regularization with differentiable parameter sensitivity modeling—eliminating reliance on ground-truth PDEs. Results: Validated on synthetic benchmarks and real biological image data (e.g., ocellated lizard skin patterning), the framework robustly localizes bifurcations in unseen parameter regimes, accurately resolves anatomically localized pattern evolution, and uncovers how geometric constraints regulate reaction–diffusion pattern formation.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Computer Vision: Other Foundations of Computer Vision

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Abstract

Spatiotemporal dynamics pervade the natural sciences, from the morphogen dynamics underlying patterning in animal pigmentation to the protein waves controlling cell division. A central challenge lies in understanding how controllable parameters induce qualitative changes in system behavior called bifurcations. This endeavor is particularly difficult in realistic settings where governing partial differential equations (PDEs) are unknown and data is limited and noisy. To address this challenge, we propose TRENDy (Temporal Regression of Effective Nonlinear Dynamics), an equation-free approach to learning low-dimensional, predictive models of spatiotemporal dynamics. TRENDy first maps input data to a low-dimensional space of effective dynamics through a cascade of multiscale filtering operations. Our key insight is the recognition that these effective dynamics can be fit by a neural ordinary differential equation (NODE) having the same parameter space as the input PDE. The preceding filtering operations strongly regularize the phase space of the NODE, making TRENDy significantly more robust to noise compared to existing methods. We train TRENDy to predict the effective dynamics of synthetic and real data representing dynamics from across the physical and life sciences. We then demonstrate how we can automatically locate both Turing and Hopf bifurcations in unseen regions of parameter space. We finally apply our method to the analysis of spatial patterning of the ocellated lizard through development. We found that TRENDy's predicted effective state not only accurately predicts spatial changes over time but also identifies distinct pattern features unique to different anatomical regions, such as the tail, neck, and body--an insight that highlights the potential influence of surface geometry on reaction-diffusion mechanisms and their role in driving spatially varying pattern dynamics.

Problem

• Understanding bifurcations in spatiotemporal dynamics with unknown PDEs.
• Learning low-dimensional models from noisy, limited data.
• Analyzing spatial patterning influenced by surface geometry.

Innovation

• Uses neural ODEs for effective dynamics modeling
• Applies multiscale filtering to reduce noise impact
• Automatically detects bifurcations in parameter space