We address data-driven learning of the infinitesimal generator of stochastic diffusion processes, essential for understanding numerical simulations of natural and physical systems.

In this section, we outline the'relaxation' or'decay' of the spectral components of Each trajectory is simulated for 100000 steps. A separate testing set was generated in an identical manner but with a different random seed. The original data was obtained upon request from DE Shaw Research, and details about the simulations are available in the original publication [19]. Figures 8, 9 and 10, show conditional distributions generated by CG-SE3-ITO models and comparisons of MD with ITO simulations on the fast folders Trp-Cage, BBA, and Villin, respectively. Reference value and observables We compute observables using Markov state models.

Molecular dynamics (MD) is a broadly adopted technique to approximate such quantities.

Deep neural networks (DNNs) achieve remarkable performance on a wide range of tasks, yet their mathematical analysis remains fragmented: stability and generalization are typically studied in disparate frameworks and on a case-by-case basis. Architecturally, DNNs rely on the recursive application of parametrized functions, a mechanism that can be unstable and difficult to train, making stability a primary concern. Even when training succeeds, there are few rigorous results on how well such models generalize beyond the observed data, especially in the generative setting. In this work, we leverage the theory of stochastic Iterated Function Systems (IFS) and show that two important deep architectures can be viewed as, or canonically associated with, place-dependent IFS. This connection allows us to import results from random dynamical systems to (i) establish the existence and uniqueness of invariant measures under suitable contractivity assumptions, and (ii) derive a Wasserstein generalization bound for generative modeling. The bound naturally leads to a new training objective that directly controls the collage-type approximation error between the data distribution and its image under the learned transfer operator. We illustrate the theory on a controlled 2D example and empirically evaluate the proposed objective on standard image datasets (MNIST, CelebA, CIFAR-10).

We develop a general theory of semantic dynamics for large language models by formalizing them as Continuous State Machines (CSMs): smooth dynamical systems whose latent manifolds evolve under probabilistic transition operators. The associated transfer operator $P: L^2(M,μ) \to L^2(M,μ)$ encodes the propagation of semantic mass. Under mild regularity assumptions (compactness, ergodicity, bounded Jacobian), $P$ is compact with discrete spectrum. Within this setting, we prove the Semantic Characterization Theorem (SCT): the leading eigenfunctions of $P$ induce finitely many spectral basins of invariant meaning, each definable in an o-minimal structure over $\mathbb{R}$. Thus spectral lumpability and logical tameness coincide. This explains how discrete symbolic semantics can emerge from continuous computation: the continuous activation manifold collapses into a finite, logically interpretable ontology. We further extend the SCT to stochastic and adiabatic (time-inhomogeneous) settings, showing that slowly drifting kernels preserve compactness, spectral coherence, and basin structure.

A general idea to overcome the above problem is to perturb the system dynamics.

In this section, we outline the'relaxation' or'decay' of the spectral components of Each trajectory is simulated for 100000 steps. A separate testing set was generated in an identical manner but with a different random seed. The original data was obtained upon request from DE Shaw Research, and details about the simulations are available in the original publication [19]. Figures 8, 9 and 10, show conditional distributions generated by CG-SE3-ITO models and comparisons of MD with ITO simulations on the fast folders Trp-Cage, BBA, and Villin, respectively. Reference value and observables We compute observables using Markov state models.

Molecular dynamics (MD) is a broadly adopted technique to approximate such quantities.

We propose a randomized neural network approach called RaNNDy for learning transfer operators and their spectral decompositions from data. The weights of the hidden layers of the neural network are randomly selected and only the output layer is trained. The main advantage is that without a noticeable reduction in accuracy, this approach significantly reduces the training time and resources while avoiding common problems associated with deep learning such as sensitivity to hyperparameters and slow convergence. Additionally, the proposed framework allows us to compute a closed-form solution for the output layer which directly represents the eigenfunctions of the operator. Moreover, it is possible to estimate uncertainties associated with the computed spectral properties via ensemble learning. We present results for different dynamical operators, including Koopman and Perron-Frobenius operators, which have important applications in analyzing the behavior of complex dynamical systems, and the Schrödinger operator. The numerical examples, which highlight the strengths but also weaknesses of the proposed framework, include several stochastic dynamical systems, protein folding processes, and the quantum harmonic oscillator.