Variational neural networks find system states without solving differential equations
Variational Functional Neural Networks (vfNN)
Machine Learning
Summary
Finding the stable states of physical systems often involves solving complex differential equations. The authors introduce a new approach called variational functional neural networks (vfNN) that avoids solving these equations directly by working with an integral form of the problem instead. This method uses physics-based scales rather than complicated weights and applies special optimization techniques to handle difficult math landscapes. Their tests show vfNN can accurately find solutions for various systems, including those governed by nonlinear equations.
What this means in practice
- •For computational physics teams: Identify stable states of physical systems without numerically solving differential equations by applying vfNN to integral functional formulations.
- •For engineering simulation teams: Improve simulation workflows by replacing complex differential equation solvers with vfNN-based functional optimization for systems described by continuum functionals.
Authors
Chinmay Katke, C. Nadir Kaplan
Abstract
We present a variational functional neural network (vfNN) algorithm that optimizes physically motivated continuum functionals in integral form (e.g., Lagrangian, Hamiltonian, or free energy) to find the states of a system. That way, vfNN bypasses solving the equations of motion (a.k.a. the Euler-Lagrange equations), which are in the form of differential equations. To this end, our approach is fundamentally different from conventional physics-informed neural networks (PINN) that solve differential equations by minimizing a phenomenological loss function involving higher order derivatives and ad hoc weight factors. In contrast, vfNN merely evaluates lower order derivatives of the generating functional and replaces the weight factors with rigorous physical scales. This also fundamentally changes the loss landscape, and the desired solutions may become unstable. For this emergent nonconvex optimization problem, we develop a quasi-Newton optimization algorithm with simulated annealing that converges to the solutions of the corresponding Euler-Lagrange equations of the functional regardless of their stability properties. Our simulations using vfNN give accurate results for benchmark systems governed by linear or nonlinear ordinary differential equations.