Automated construction of effective potential via algorithmic implicit bias
Xingjie Helen Li, Molei Tao
TL;DR
The paper addresses learning effective potentials at user-selected scales for multiscale objectives by exploiting the implicit bias of large-step gradient descent, which trades small-scale structure for stochasticity. It introduces a self-learning framework that simulates a damped mechanical system at a chosen step size, estimates the unresolved-scale covariance to impose a fluctuation–dissipation balance, and recovers an effective potential $U_k$ suitable for surrogate Hamiltonian or Langevin models. The approach is validated through one- and two-dimensional numerical experiments, including anisotropic noise calibration and multi-well potentials, demonstrating accurate reproduction of equilibrium statistics, mean paths, and auto-correlation functions under the surrogate dynamics. This provides a scalable method to construct reduced-order, scale-aware models with practical impact in accelerating simulations and improving interpretability in multiscale physical and data-driven problems.
Abstract
We introduce a novel approach for decomposing and learning every scale of a given multiscale objective function in $\mathbb{R}^d$, where $d\ge 1$. This approach leverages a recently demonstrated implicit bias of the optimization method of gradient descent by Kong and Tao, which enables the automatic generation of data that nearly follow Gibbs distribution with an effective potential at any desired scale. One application of this automated effective potential modeling is to construct reduced-order models. For instance, a deterministic surrogate Hamiltonian model can be developed to substantially soften the stiffness that bottlenecks the simulation, while maintaining the accuracy of phase portraits at the scale of interest. Similarly, a stochastic surrogate model can be constructed at a desired scale, such that both its equilibrium and out-of-equilibrium behaviors (characterized by auto-correlation function and mean path) align with those of a damped mechanical system with the original multiscale function being its potential. The robustness and efficiency of our proposed approach in multi-dimensional scenarios have been demonstrated through a series of numerical experiments. A by-product of our development is a method for anisotropic noise estimation and calibration. More precisely, Langevin model of stochastic mechanical systems may not have isotropic noise in practice, and we provide a systematic algorithm to quantify its covariance matrix without directly measuring the noise. In this case, the system may not admit closed form expression of its invariant distribution either, but with this tool, we can design friction matrix appropriately to calibrate the system so that its invariant distribution has a closed form expression of Gibbs.