Sparse identification of quasipotentials via a combined data-driven method

TL;DR

This work combines two existing data-driven techniques, namely a neural network and a sparse regression algorithm, specifically designed to symbolically describe multistable energy landscapes, to demonstrate how to discover parsimonious equations for the quasipotential directly from data.

Abstract

The quasipotential function allows for comprehension and prediction of the escape mechanisms from metastable states in nonlinear dynamical systems. This function acts as a natural extension of the potential function for non-gradient systems and it unveils important properties such as the maximum likelihood transition paths, transition rates and expected exit times of the system. Here, we demonstrate how to discover parsimonious equations for the quasipotential directly from data. Leveraging machine learning, we combine two existing data-driven techniques, namely a neural network and a sparse regression algorithm, specifically designed to symbolically describe multistable energy landscapes. First, we employ a vanilla neural network enhanced with a renormalization and rescaling procedure to achieve an orthogonal decomposition of the vector field. Next, we apply symbolic regression to extract the downhill and circulatory components of the decomposition, ensuring consistency with the underlying dynamics. This symbolic reconstruction involves a simultaneous regression that imposes constraints on both the orthogonality condition and the vector field. We implement and benchmark our approach using an archetypal model with a known exact quasipotential, as well as a nanomechanical resonator system. We further demonstrate its applicability to noisy data and to a four-dimensional system. Our model-unbiased analytical forms of the quasipotential is of interest to a wide range of applications aimed at assessing metastability and energy landscapes, serving to parametrically capture the distinctive fingerprint of the fluctuating dynamics.

Figures (11)

• Figure 1: Combined data-driven method applied to the system \ref{['eq_modelsystem1']} with $5000$ sampled trajectories. (a) Five generated trajectories. (b) The identified downhill component $-\nabla V^{\text{Symb}}(x,y,z)$. (c) The identified circulatory component $\mathbf{g}^{\text{Symb}}(x,y,z)$. (d) The exact quasipotential $U(x,y,z)$. (e) The identified quasipotential $U^{\text{Symb}}(x,y,z)$. All plots are projected on the $(x,y)$-plane. In panels (b) and (c), the line thickness shows the flow velocity.
• Figure 2: Distributions of the coefficients for each library term in the identified potential $U^{\text{Symb}}(\mathbf{x})$ in $50$ independent runs for the system in Eq. \ref{['eq_modelsystem1']}. Panel (a)-(e) indicate the coefficient distributions for the nontrivial terms ($1$, $x^2$, $y^2$, $z^2$ and $x^4$) as in $U(\mathbf{x})$, while panel (f) shows that learned coefficients are zero in all runs for any of the remaining library terms.
• Figure 3: Error of the symbolic model with respect the exact one for the 50 repetitions of the data-driven symbolic identification.
• Figure 4: (a) Phase plot of six observed trajectories and the trajectories generated by simulating the dynamics $\dot{\mathbf{x}}=\mathbf{f}^{\text{Symb}}(\mathbf{x})$ with the same initial states. (b)-(d) Plots of the invariant distribution computed by $p_{\epsilon}(x,y,z)=Z_{\epsilon}^{-1}\exp(-U^{\text{Symb}}(x,y,z)/\epsilon)$ of the randomly perturbed dynamics system \ref{['SDE']} with various values of $\epsilon$ ($0.05$, $0.1$ and $0.2$). All plots are projected on the $(x,y)$-plane.
• Figure 5: Combined data-driven method applied to the system \ref{['eq_modelsystem1_noise']} with $5000$ sampled noisy trajectories. (a) Six generated trajectories and the ones sampled from the deterministic system \ref{['eq_modelsystem1']} with the same initial conditions. (b) Six trajectories generated by simulating the learned dynamics $\dot{\mathbf{x}}=\mathbf{f}^{\text{Symb}}(\mathbf{x})$ and the ones sampled from the system \ref{['eq_modelsystem1']} with the same initial states. (c) The identified quasipotential $U^{\text{Symb}}(x,y,z)$. All plots are projected on the $(x,y)$-plane.