Data-driven Evolutions of Critical Points
This addresses the challenge of energy reconstruction in physics and engineering from limited observational data, though it is incremental as it builds on existing variational and convergence frameworks.
The paper tackles the problem of learning energies from data by observing time evolutions of critical points, proving convergence to the exact energy via Gamma-convergence and demonstrating it with analytic and numerical results for a one-dimensional nonlinear-elastic rod model.
In this paper we are concerned with the learnability of energies from data obtained by observing time evolutions of their critical points starting at random initial equilibria. As a byproduct of our theoretical framework we introduce the novel concept of mean-field limit of critical point evolutions and of their energy balance as a new form of transport. We formulate the energy learning as a variational problem, minimizing the discrepancy of energy competitors from fulfilling the equilibrium condition along any trajectory of critical points originated at random initial equilibria. By Gamma-convergence arguments we prove the convergence of minimal solutions obtained from finite number of observations to the exact energy in a suitable sense. The abstract framework is actually fully constructive and numerically implementable. Hence, the approximation of the energy from a finite number of observations of past evolutions allows to simulate further evolutions, which are fully data-driven. As we aim at a precise quantitative analysis, and to provide concrete examples of tractable solutions, we present analytic and numerical results on the reconstruction of an elastic energy for a one-dimensional model of thin nonlinear-elastic rod.