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Predicting the Energy Landscape of Stochastic Dynamical System via Physics-informed Self-supervised Learning

Ruikun Li, Huandong Wang, Qingmin Liao, Yong Li

TL;DR

PESLA addresses the challenge of inferring energy landscapes from stochastic dynamics without direct energy supervision by combining adaptive codebook-based discretization with a Graph Neural Fokker-Planck equation to capture energy-driven drift and diffusion on a discretized landscape. It introduces a physics-informed regularization that biases long-term dynamics toward Boltzmann-like behavior, enabling accurate energy estimation (correlations exceeding $0.9$) and improved evolution prediction (about $17.65\%$ gains over baselines) across diverse domains including 2D Prinz potentials, ecological evolution, and protein folding. The approach reduces reliance on costly energy annotations, demonstrates robustness to limited data and noise, and shows transferable components across related systems, making it broadly applicable to multidisciplinary stochastic dynamics.

Abstract

Energy landscapes play a crucial role in shaping dynamics of many real-world complex systems. System evolution is often modeled as particles moving on a landscape under the combined effect of energy-driven drift and noise-induced diffusion, where the energy governs the long-term motion of the particles. Estimating the energy landscape of a system has been a longstanding interdisciplinary challenge, hindered by the high operational costs or the difficulty of obtaining supervisory signals. Therefore, the question of how to infer the energy landscape in the absence of true energy values is critical. In this paper, we propose a physics-informed self-supervised learning method to learn the energy landscape from the evolution trajectories of the system. It first maps the system state from the observation space to a discrete landscape space by an adaptive codebook, and then explicitly integrates energy into the graph neural Fokker-Planck equation, enabling the joint learning of energy estimation and evolution prediction. Experimental results across interdisciplinary systems demonstrate that our estimated energy has a correlation coefficient above 0.9 with the ground truth, and evolution prediction accuracy exceeds the baseline by an average of 17.65\%. The code is available at github.com/tsinghua-fib-lab/PESLA.

Predicting the Energy Landscape of Stochastic Dynamical System via Physics-informed Self-supervised Learning

TL;DR

PESLA addresses the challenge of inferring energy landscapes from stochastic dynamics without direct energy supervision by combining adaptive codebook-based discretization with a Graph Neural Fokker-Planck equation to capture energy-driven drift and diffusion on a discretized landscape. It introduces a physics-informed regularization that biases long-term dynamics toward Boltzmann-like behavior, enabling accurate energy estimation (correlations exceeding ) and improved evolution prediction (about gains over baselines) across diverse domains including 2D Prinz potentials, ecological evolution, and protein folding. The approach reduces reliance on costly energy annotations, demonstrates robustness to limited data and noise, and shows transferable components across related systems, making it broadly applicable to multidisciplinary stochastic dynamics.

Abstract

Energy landscapes play a crucial role in shaping dynamics of many real-world complex systems. System evolution is often modeled as particles moving on a landscape under the combined effect of energy-driven drift and noise-induced diffusion, where the energy governs the long-term motion of the particles. Estimating the energy landscape of a system has been a longstanding interdisciplinary challenge, hindered by the high operational costs or the difficulty of obtaining supervisory signals. Therefore, the question of how to infer the energy landscape in the absence of true energy values is critical. In this paper, we propose a physics-informed self-supervised learning method to learn the energy landscape from the evolution trajectories of the system. It first maps the system state from the observation space to a discrete landscape space by an adaptive codebook, and then explicitly integrates energy into the graph neural Fokker-Planck equation, enabling the joint learning of energy estimation and evolution prediction. Experimental results across interdisciplinary systems demonstrate that our estimated energy has a correlation coefficient above 0.9 with the ground truth, and evolution prediction accuracy exceeds the baseline by an average of 17.65\%. The code is available at github.com/tsinghua-fib-lab/PESLA.

Paper Structure

This paper contains 32 sections, 6 equations, 15 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Background and problem setup
  3. Method
  4. Adaptive Codebook Learning
  5. Graph Neural Fokker-Planck Equation
  6. Training
  7. Experiments
  8. Setup
  9. 2D Prinz Potential
  10. Ecological Evolution
  11. Protein Folding
  12. Related Work
  13. Energy Estimation
  14. Evolution Prediction
  15. Conclusion
  16. ...and 17 more sections

Figures (15)

  • Figure 1: Framework of PESLA. (a) The energy landscape with evolution trajectories; (b) Partitioning the state space with an adaptive codebook to form the codewords with a graph topology and modeling the time evolution of probability across neighboring regions by graph neural Fokker-Planck equation.
  • Figure 2:
  • Figure 3:
  • Figure 5:
  • Figure 7: Robustness
  • ...and 10 more figures