Learning invariant representations of time-homogeneous stochastic dynamical systems

TL;DR

The paper addresses learning invariant representations for time-homogeneous stochastic dynamics to enable accurate estimation of the transfer operator $\mathcal{T}$ or generator $\mathcal{L}$. It introduces Deep Projection Networks (DPNets) that optimize projection-based scores $\mathcal{P}$ and $\mathcal{S}^\gamma$ to identify leading invariant subspaces, with a differentiable relaxation that improves numerical stability and a continuous-time extension for time-reversal invariant systems. Theoretical results show that these scores capture the leading invariant subspace under compactness (and in the continuous case under self-adjointness) and link representation learning with operator regression (EDMD) for forecasting and spectral analysis. Extensive experiments across logistic maps, MNIST-like dynamics, fluid flow, and molecular systems demonstrate robustness and scalability, highlighting the practical impact for data-driven dynamical systems.

Abstract

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

Figures (9)

• Figure 1: Pipeline for learning dynamical systems. DPNets learn a data representation to be used with standard operator regression methods. In turn, these are employed to solve downstream tasks such as forecasting and interpreting dynamical systems via spectral decomposition.
• Figure 2: Eigenvalue error decays during training. Upper: spectral error for the logistic map. Lower: DPNets-estimated eigenvalues of $\mathcal{L}$ for the Langevin dynamics.
• Figure 3: Forecasting with DPNets. Upper: classification accuracy over time for ordered MNIST. Lower: forecasting RMSE over time for fluid dynamic example.
• Figure 4: Free energy surface of the 2 slowest modes of Chignolin,estimated by DPNets and Nystr̈om PCR. To be compared with Bonati2021.
• Figure 5: (Left) Decay of the singular values $\sigma_{i}(\mathcal{T})$ and of the the eigenvalues $|\lambda_{i}(\mathcal{T})|$ for the logistic map example. (Center, Right) Spectral error and Optimality gap as a function of the feature dimension $r$ for the baselines considered in Tab. \ref{['tab:1']}.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Definition 1: Markov process.
• Definition 2
• Definition 3: Transfer operator
• Definition 4: Feller process
• Definition 5: Infinitesimal generator of a semigroup
• Lemma 1
• proof
• Remark 1