Discovering Latent Response Laws in Forced Physical Systems

2026-07-09

AI Summary

In high-dimensional observations, physical systems driven by external time-varying inputs often obscure their underlying response laws due to strong coupling between effective states and inputs, posing significant challenges for modeling and prediction. This work proposes FLARE, the first method to extend equation discovery to forced dynamical systems. FLARE employs a forced latent-variable autoencoder to learn compact response coordinates that disentangle system states from inputs, and combines sparse regression to identify sparse, input-dependent dynamics in the latent space. Through joint optimization with the decoder, it enables interpretable dynamic modeling. Requiring only historical response data for initialization, FLARE accurately predicts long-term, high-dimensional responses under unseen input conditions, successfully recovering compact forced dynamics across canonical systems, application-scale forced tasks, and visual observations.

Abstract

Governing equations provide compact descriptions of physical systems, yet the variables in which they are simple are often hidden in high-dimensional measurements. This challenge is sharper for forced systems, whose responses depend on both intrinsic dynamics and time-dependent inputs. Here we introduce FLARE, a forced latent autoencoder for response equations that learns compact response coordinates, identifies sparse input-dependent latent dynamics and decodes equation rollouts to full responses. By estimating latent dimension from data and separating state estimation from external forcing, FLARE enables forecasts to be initialized from past responses and driven by prescribed future inputs. Across known dynamical systems, application-scale forced responses and visual observations, FLARE recovers compact forced dynamics and predicts long-horizon high-dimensional responses under inputs not used for training. By turning learned coordinates into a dynamical interface, FLARE extends equation discovery to systems whose effective states are hidden within complex observations, providing a route for interpretable modelling and prediction of high-dimensional responses in forced dynamical systems.

Problem

• forced systems
• latent dynamics
• equation discovery
• high-dimensional responses
• response laws

Innovation

• forced dynamical systems
• latent dynamics
• equation discovery
• autoencoder
• sparse identification