EPR-Net: Constructing non-equilibrium potential landscape via a variational force projection formulation

TL;DR

EPR-Net is presented, a novel and effective deep learning approach that tackles a crucial challenge in biophysics: constructing potential landscapes for high-dimensional non-equilibrium steady-state systems and offers a promising solution for diverse landscape construction problems in biophysics.

Abstract

We present EPR-Net, a novel and effective deep learning approach that tackles a crucial challenge in biophysics: constructing potential landscapes for high-dimensional non-equilibrium steady-state (NESS) systems. EPR-Net leverages a nice mathematical fact that the desired negative potential gradient is simply the orthogonal projection of the driving force of the underlying dynamics in a weighted inner-product space. Remarkably, our loss function has an intimate connection with the steady entropy production rate (EPR), enabling simultaneous landscape construction and EPR estimation. We introduce an enhanced learning strategy for systems with small noise, and extend our framework to include dimensionality reduction and state-dependent diffusion coefficient case in a unified fashion. Comparative evaluations on benchmark problems demonstrate the superior accuracy, effectiveness, and robustness of EPR-Net compared to existing methods. We apply our approach to challenging biophysical problems, such as an 8D limit cycle and a 52D multi-stability problem, which provide accurate solutions and interesting insights on constructed landscapes. With its versatility and power, EPR-Net offers a promising solution for diverse landscape construction problems in biophysics.

Figures (8)

• Figure 1: Constructing energy landscapes through enhanced EPR workflow. (a) The primary objective is to construct the energy landscape defined through the steady-state distribution of the system. (b) Constructing the high-dimensional energy landscape using the EPR framework with primitive variables. (c) Constructing the dimensionality-reduced energy landscape using EPR with prescribed reduced variables.
• Figure 2: Two-dimensional benchmark examples solved under the EPR framework. (a-c) Filled contour plots of the learned potential $V(\boldsymbol x;\theta^*)$ for (a) a toy model learned by the EPR loss (\ref{['eq:L-EPR']}), (b) a biochemical oscillation network model Wang08 and (c) a tri-stable cell development model Wang11, all of which are learned by the enhanced loss (\ref{['eq:enh']}). The force field $\boldsymbol F(\boldsymbol x)$ is decomposed into the gradient part $-\nabla V(\boldsymbol x;\theta^*)$ (white arrows) and the non-gradient part $\boldsymbol F(\boldsymbol x)+\nabla V(\boldsymbol x;\theta^*)$ (gray arrows). The length of an arrow corresponds to the magnitude of the vector. The solid dots are samples from the simulated invariant distribution. (d-f) SDE-simulated samples $(\boldsymbol x_i)_{1\le i\le N}$ (yellow points) and enhanced samples $(\boldsymbol x'_i)_{1\le i\le N'}$ (green points). (d) $D=0.1$, where enhanced samples are generated with $D'=2D$. (e) $D=0.1$, where enhanced samples are obtained by adding Gaussian perturbations with $\sigma=0.05$ on SDE-simulated samples. (f) $D=0.01$, where enhanced samples are generated with $D'=10D$.
• Figure 3: Comparisons between models learned by (a,d) enhanced EPR, (b,e) HJB loss alone and (c,f) normalizing flow. (a-c) Filled contour plots of the potential $V(\boldsymbol x;\theta^*)$ for the toy model with $D=0.05$. The force field $\boldsymbol F(\boldsymbol x)$ is decomposed into the gradient part $-\nabla V(\boldsymbol x;\theta^*)$ (white arrows) and the non-gradient part (gray arrows). The length of an arrow denotes the scale of the vector. The solid dots are samples from the simulated invariant distribution. The results in the high-energy region $\{\boldsymbol x | V(\boldsymbol x) \geq 30D\}$ are omitted since they are less relevant to the dynamics. (d-f) The absolute error of the learned potential constructed in different ways.
• Figure 4: Landscapes constructed by enhanced EPR. (a) Slices of the learned 3D potential ${V}(\boldsymbol x;\theta^*)$ in the Lorenz system. The solid dots are samples from the simulated invariant distribution. (b) The accumulated measure of the absolute error between $U_0(\boldsymbol x)$ and the learned solution $V(\boldsymbol x; \theta^*)$ based on the conditional probability $p_0(\boldsymbol x |x_1, x_2)$. (c) The learned 12D potential $V(\boldsymbol x; \theta^*)$ along a line $\boldsymbol x(t)=t \boldsymbol \mu_1 + (1-t) \boldsymbol \mu_2$, where $\boldsymbol \mu_1,\boldsymbol \mu_2$ are the means of two components in a 12D Gaussian mixture.
• Figure 5: Dimensionality reduction of high-dimensional systems. (a-c) Eight-dimensional cell cycle model. (a) Reduced potential landscape $\widetilde{V}$ with projected contour lines. The green star at $(0.13,0.59)$ denotes the saddle point of $\widetilde{V}$. (b) Projected sample points, streamlines of the projected force field $\widetilde{\boldsymbol G}(\boldsymbol x;\theta^*)$ and the filled contour plot of $\widetilde{V}(\boldsymbol x;\theta^*)$. Two red circles and two red dots (close to $(0.22,0.11)$ and $(0.31,0.47)$, respectively) show the stable limit sets of the projected force field. The yellow circle is the projection of the original high-dimensional limit cycle. (c) An enlarged view of the square domain in (b), showing the detailed spiral structure of the streamlines of $\widetilde{\boldsymbol G}(\boldsymbol x;\theta^*)$ around the stable point. (d and e) Fifty-two-dimensional multi-stable system. (d) Projected force $\widetilde{\boldsymbol G}(\boldsymbol z;\theta^*)$ and potential $\widetilde{V_1}(\boldsymbol z;\theta^*)$ of the 52D double-well model learned by enhanced EPR. (e) The absolute error of the reduced potential constructed in different ways, i.e., $|\widetilde{V_1}(\boldsymbol z;\theta^*) - \widetilde{V_2}(\boldsymbol z;\theta^*)|$.