New neural network models systems with multiple stable states
Port-Hamiltonian Neural Networks for Systems with Multiple Asymptotically Stable Equilibria
Machine Learning
Summary
Some physical or engineered systems can settle into different stable conditions, but existing neural network models only handle one. The authors find that current methods miss system behaviors like two stable points and fix it by designing neural networks that represent multiple valleys of energy. They prove their approach is mathematically sound, showing all stable states stay within a certain range and that the system behaves well in most cases. Their method works better than previous ones on example problems, making predictions converge faster.
What this means in practice
- •For control systems engineers: Model and simulate systems with multiple stable operating points more accurately to improve controller design and safety verification.
- •For roboticists: Predict robot dynamics exhibiting multiple stable poses or configurations for smoother motion planning and state estimation.
Authors
Simon Heilig, Jens Püttschneider, Mohammad Itani, Asja Fischer, Timm Faulwasser
Abstract
Stable port-Hamiltonian neural networks certify asymptotic stability by construction. Yet, their Hamiltonian is a global Lyapunov function with a single global minimum, so they can represent only dynamic systems with {one} attractor. We demonstrate that this excludes even simple systems with energy landscapes forming a double well, and we overcome the restriction by parametrising the Hamiltonian as a {product} of Bregman divergences generated by one input-convex network. We prove that the resulting model is locally Lyapunov stable, that the coexistence of stable equilibria forces additional non-asymptotically-stable equilibria to exist, that all equilibria lie in a bounded region, and under a hyperbolicity assumption that almost-everywhere stability holds. On three systems our approach is able to recover the energy surface characteristics and improve the convergence speed by 1.8$\times$-8.5$\times$.