[Paper Review] A Data Driven Method for Computing Quasipotentials

Bo Lin, Qianxiao Li+1 more|arXiv (Cornell University)|Dec 13, 2020

Model Reduction and Neural Networks26 references6 citations

TL;DR

This paper proposes a data-driven machine learning method to compute quasipotentials in high-dimensional stochastic dynamical systems without requiring explicit models. By learning an orthogonal decomposition of the vector field into potential and rotational components via neural networks trained on trajectory data, the method efficiently reconstructs the quasipotential landscape, enabling accurate analysis of rare events and transition paths in complex systems such as high-dimensional PDEs and biological networks.

ABSTRACT

The quasipotential is a natural generalization of the concept of energy functions to non-equilibrium systems. In the analysis of rare events in stochastic dynamics, it plays a central role in characterizing the statistics of transition events and the likely transition paths. However, computing the quasipotential is challenging, especially in high dimensional dynamical systems where a global landscape is sought. Traditional methods based on the dynamic programming principle or path space minimization tend to suffer from the curse of dimensionality. In this paper, we propose a simple and efficient machine learning method to resolve this problem. The key idea is to learn an orthogonal decomposition of the vector field that drives the dynamics, from which one can identify the quasipotential. We demonstrate on various example systems that our method can effectively compute quasipotential landscapes without requiring spatial discretization or solving path-space optimization problems. Moreover, the method is purely data driven in the sense that only observed trajectories of the dynamics are required for the computation of the quasipotential. These properties make it a promising method to enable the general application of quasipotential analysis to dynamical systems away from equilibrium.

Motivation & Objective

• Address the challenge of computing quasipotentials in high-dimensional non-equilibrium dynamical systems where traditional methods fail due to the curse of dimensionality.
• Overcome limitations of path-space minimization (limited to specific transition points) and mesh-based dynamic programming (infeasible in high dimensions).
• Develop a purely data-driven approach that learns the quasipotential landscape directly from observed trajectories without requiring the underlying dynamical model.
• Enable scalable, global quasipotential mapping for systems with complex attractors, such as stable equilibria, limit cycles, and high-dimensional spatiotemporal dynamics.
• Facilitate downstream analysis of rare events, minimum action paths, and exit times from basins of attraction using the computed quasipotential landscape.

Proposed method

• Parameterize the vector field as the sum of a potential component (gradient of a scalar potential function) and a rotational component (divergence-free vector field), both modeled by deep neural networks.
• Train the neural networks using a loss function that combines two components: (1) reconstruction error of the observed trajectories, and (2) enforcement of orthogonality between the potential and rotational components.
• Enforce orthogonality via a constraint that the inner product of the potential and rotational vector fields vanishes at sampled data points, ensuring physical consistency.
• Compute the quasipotential by restricting the learned potential component to the basin of attraction of each attractor, using the value at the attractor as a reference point.
• Use the learned dynamics to simulate trajectories and validate the accuracy of the quasipotential landscape against known benchmarks such as the minimum action method.
• Apply the method to high-dimensional systems, including a 40-dimensional reaction-diffusion system, by projecting the state onto a low-dimensional subspace defined by dominant Fourier modes.

Experimental results

Research questions

• RQ1 Can a data-driven machine learning method accurately compute the global quasipotential landscape in high-dimensional stochastic dynamical systems without explicit model knowledge?
• RQ2 How well can neural networks learn the orthogonal decomposition of the vector field into potential and rotational components from trajectory data alone?
• RQ3 Can the method effectively capture the quasipotential landscape for systems with complex attractors, such as limit cycles and high-dimensional spatiotemporal dynamics?
• RQ4 What is the accuracy and convergence behavior of the method compared to established path-space minimization and mesh-based dynamic programming approaches?
• RQ5 To what extent can the computed quasipotential be used to predict rare transition events, minimum action paths, and exit times from basins of attraction?

Key findings

• The method successfully computes the quasipotential landscape for a 40-dimensional reaction-diffusion system using only $4 \times 10^6$ trajectory data points, with a reconstruction error of $8.227 \times 10^{-4} \pm 6.741 \times 10^{-4}$ on test trajectories.
• The quasipotential contours computed via the data-driven method agree well with those obtained using the minimum action method, validating its accuracy.
• The method achieves global quasipotential mapping without spatial discretization or path optimization, overcoming the scalability limitations of traditional approaches.
• The neural network-based orthogonal decomposition effectively separates the vector field into physically meaningful potential and rotational components, enabling robust quasipotential estimation.
• The approach enables downstream analysis such as identifying minimum action paths and estimating exit times from basins of attraction using the quasipotential landscape.
• The method is purely data-driven: it learns the underlying dynamics and quasipotential simultaneously from observed trajectories, without requiring the explicit form of the stochastic differential equation.