Recurrent neural networks (RNNs) notoriously struggle to learn long-term memories, primarily due to vanishing and exploding gradients.

Neural Networks (GNN)) to perform the reparametrization, incorporating CG modes as needed.

The derivation is based on Section 3.2.

But it can also be problematic since it can induce inconsistencies in, e.g., Hessian-based flatness measures,

Current approximate posteriors in Bayesian neural networks (BNNs) exhibit a crucial limitation: they fail to maintain invariance under reparameterization, i.e.

We propose a novel approach to molecular simulations using neural network reparametrization, which offers a flexible alternative to traditional coarse-graining methods. Unlike conventional techniques that strictly reduce degrees of freedom, the complexity of the system can be adjusted in our model, sometimes increasing it to simplify the optimization process. Our approach also maintains continuous access to fine-grained modes and eliminates the need for force-matching, enhancing both the efficiency and accuracy of energy minimization.Importantly, our framework allows for the use of potentially arbitrary neural networks (e.g., Graph Neural Networks (GNN)) to perform the reparametrization, incorporating CG modes as needed. In fact, our experiments using very weak molecular forces (Lennard-Jones potential) the GNN-based model is the sole model to find the correct configuration. Similarly, in protein-folding scenarios, our GNN-based CG method consistently outperforms traditional optimization methods. It not only recovers the target structures more accurately but also achieves faster convergence to the deepest energy states.This work demonstrates significant advancements in molecular simulations by optimizing energy minimization and convergence speeds, offering a new, efficient framework for simulating complex molecular systems.

Model reparametrization, which follows the change-of-variable rule of calculus, is a popular way to improve the training of neural nets. But it can also be problematic since it can induce inconsistencies in, e.g., Hessian-based flatness measures, optimization trajectories, and modes of probability densities. This complicates downstream analyses: e.g. one cannot definitively relate flatness with generalization since arbitrary reparametrization changes their relationship. In this work, we study the invariance of neural nets under reparametrization from the perspective of Riemannian geometry. From this point of view, invariance is an inherent property of any neural net if one explicitly represents the metric and uses the correct associated transformation rules. This is important since although the metric is always present, it is often implicitly assumed as identity, and thus dropped from the notation, then lost under reparametrization. We discuss implications for measuring the flatness of minima, optimization, and for probability-density maximization. Finally, we explore some interesting directions where invariance is useful.