  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.
