Rapid training of Hamiltonian graph networks without gradient descent

Learning dynamical systems that respect physical symmetries and constraints remains a fundamental challenge in data-driven modeling. Integrating physical laws with graph neural networks facilitates principled modeling of complex N-body dynamics and yields accurate and permutation-invariant models. However, training graph neural networks with iterative, gradient-based optimization algorithms (e.g., Adam, RMSProp, LBFGS) often leads to slow training, especially for large, complex systems. In comparison to 15 different optimizers, we demonstrate that Hamiltonian Graph Networks (HGN) can be trained up to 600x faster--but with comparable accuracy--by replacing iterative optimization with random feature-based parameter construction. We show robust performance in diverse simulations, including N-body mass-spring systems in up to 3 dimensions with different geometries, while retaining essential physical invariances with respect to permutation, rotation, and translation. We reveal that even when trained on minimal 8-node systems, the model can generalize in a zero-shot manner to systems as large as 4096 nodes without retraining. Our work challenges the dominance of iterative gradient-descent-based optimization algorithms for training neural network models for physical systems.

JaxSGMC: Modular stochastic gradient MCMC in JAX

We present JaxSGMC, an application-agnostic library for stochastic gradient Markov chain Monte Carlo (SG-MCMC) in JAX. SG-MCMC schemes are uncertainty quantification (UQ) methods that scale to large datasets and high-dimensional models, enabling trustworthy neural network predictions via Bayesian deep learning. JaxSGMC implements several state-of-the-art SG-MCMC samplers to promote UQ in deep learning by reducing the barriers of entry for switching from stochastic optimization to SG-MCMC sampling. Additionally, JaxSGMC allows users to build custom samplers from standard SG-MCMC building blocks. Due to this modular structure, we anticipate that JaxSGMC will accelerate research into novel SG-MCMC schemes and facilitate their application across a broad range of domains.

Gradient-free training of recurrent neural networks

Recurrent neural networks are a successful neural architecture for many time-dependent problems, including time series analysis, forecasting, and modeling of dynamical systems. Training such networks with backpropagation through time is a notoriously difficult problem because their loss gradients tend to explode or vanish. In this contribution, we introduce a computational approach to construct all weights and biases of a recurrent neural network without using gradient-based methods. The approach is based on a combination of random feature networks and Koopman operator theory for dynamical systems. The hidden parameters of a single recurrent block are sampled at random, while the outer weights are constructed using extended dynamic mode decomposition. This approach alleviates all problems with backpropagation commonly related to recurrent networks. The connection to Koopman operator theory also allows us to start using results in this area to analyze recurrent neural networks. In computational experiments on time series, forecasting for chaotic dynamical systems, and control problems, as well as on weather data, we observe that the training time and forecasting accuracy of the recurrent neural networks we construct are improved when compared to commonly used gradient-based methods.