Solving Functional Optimization with Deep Networks and Variational Principles

TL;DR

This work tackles functional optimization with moving boundaries, where terminal time and state are unknown, by embedding the calculus of variations into a neural network framework. CalVNet learns state, control, and costate trajectories (and possibly the terminal time) without ground-truth data by minimizing $\Psi=\sum_i \|\psi_i\|^2$ to enforce variational optimality, i.e., $\delta\mathcal{J}=0$. The authors validate on three canonical problems: reproducing the Kalman filter in a variational setting, deriving bang-bang minimum-time control, and computing geodesics on manifolds, obtaining results that match analytical solutions and demonstrating robustness beyond fixed horizons. This principled, unsupervised approach offers a data-efficient pathway to solve complex functional optimization problems where traditional methods struggle due to moving boundaries or unknown supports.

Abstract

Can neural networks solve math problems using first a principle alone? This paper shows how to leverage the fundamental theorem of the calculus of variations to design deep neural networks to solve functional optimization without requiring training data (e.g., ground-truth optimal solutions). Our approach is particularly crucial when the solution is a function defined over an unknown interval or support\textemdash such as in minimum-time control problems. By incorporating the necessary conditions satisfied by the optimal function solution, as derived from the calculus of variation, in the design of the deep architecture, CalVNet leverages overparameterized neural networks to learn these optimal functions directly. We validate CalVNet by showing that, without relying on ground-truth data and simply incorporating first principles, it successfully derives the Kalman filter for linear filtering, the bang-bang optimal control for minimum-time problems, and finds geodesics on manifolds. Our results demonstrate that CalVNet can be trained in an unsupervised manner, without relying on ground-truth data, establishing a promising framework for addressing general, potentially unsolved functional optimization problems that still lack analytical solutions.

Figures (4)

• Figure 1: CalVNet learns the Kalman filter, deriving the optimal value of the functional cost. The baseline NN performs well in the time interval where ground truth is available, but fails to learn the optimal steady-state Kalman gain $G^\star_\infty$, resulting in diverging error. The baseline PINN shows diverging error. CalVNet learns the optimal steady-state error covariance $\Sigma^\star_\infty$ and Kalman gain $G^\star_\infty$ and they remain convergent beyond the time interval of the problem $[0,5]$
• Figure 2: CalVNet generates the trajectory of the state, the costate, and the control over the time interval of interest that matches the optimal trajectory. Most importantly, CalVNet learns the bang-bang behavior where control $u$ is a negative sign function of $\lambda_2$ and correctly learns the minimum time $t^\star_f$.
• Figure 3: CalVNet finds the geodesics path from the point $p_0$ to the equator. The optimal path is given by a geodesics that past through the north pole and the point $p_0$
• Figure 4: CalVNet finds the geodesics path on hyperbolic paraboloid from point $p_0$ to $p_1$