Neural Kolmogorov Equations: Parallelizable Learning of Stochastic Dynamics under General Noise

Hybrid Analytical Numerical and Machine Learning Frameworks for Solving Deterministic and Stochastic Differential Equations with Stability, Convergence, and Uncertainty Quantification

Differential equations underpin the modeling of dynamical systems across physics, engineering, biology, and finance. While deterministic ordinary and partial differential equations (ODEs/PDEs) describe systems governed by known physical laws, stochastic differential equations (SDEs) incorporate randomness to represent uncertainty, noise, and unresolved scales. Classical analytical methods are limited to restricted problem classes, and although numerical discretization provide general applicability, they can become computationally demanding for nonlinear, high-dimensional, stiff, or multiscale systems. In stochastic settings, accurate estimation further requires large ensembles of sample paths, amplifying computational cost. Conversely, purely data-driven machine learning (ML) approaches offer expressive approximation capabilities but often lack physical consistency, stability guarantees, and reliable generalization. This work introduces a unified hybrid analytical–numerical–ML framework for deterministic and stochastic differential equations. The approach integrates (i) analytical structure and prior knowledge (e.g., conservation laws and invariants), (ii) stable numerical discretization’s serving as computational backbones and multi-fidelity supervision sources, and (iii) physics-guided learning components acting as correction operators or drift–diffusion estimators. Governing equations and boundary/initial conditions are embedded directly into the learning objective, while stability constraints are enforced to preserve numerical robustness. An explicit error decomposition separates discretization, sampling, optimization, and generalization contributions, and sufficient conditions for stable and convergent hybrid approximations are derived. Numerical experiments on representative PDE and SDE benchmarks demonstrate improved accuracy and stability over backbone-only and ML-only baselines. The proposed framework provides a principled pathway toward physically consistent, scalable, and uncertainty-aware solvers for complex dynamical systems.

StocBench: A Benchmark for Generative Modeling of Stochastic Dynamics

We benchmark transport-based generative models as well as distillation-based few-step methods for the probabilistic forecasting of stochastic fluid flows, with a particular focus on performance under limited inference budgets. All methods are evaluated on a two-dimensional Kolmogorov flow with stochastic forcing. We measure one-step distributional accuracy against large simulated reference ensembles and assess whether the invariant measure is preserved during autoregressive rollouts via the enstrophy spectrum. On the stochastic task, flow matching achieves the most accurate one-step conditional distribution at high inference budgets, while the second-order exponential integrator DPM-2 is strongest at very low NFE. Few-step distillation methods are competitive with the multi-step methods and preserve the enstrophy spectrum particularly well. A deterministic control task, in which the forcing over the prediction interval is observed, separates aleatoric from epistemic uncertainty. Model performance does not translate between the two settings: the distilled models are competitive on the stochastic task but least accurate on the control task. While stochastic diffusion samplers such as DDPM better preserve the enstrophy spectrum during rollouts in the stochastic setting, deterministic samplers such as DDIM and DPM-2 show better spectral preservation in the deterministic setting.

Advances in Neural Controlled Differential Equations

This thesis advances the training and scalability of NCDEs through three complementary contributions, building on neural rough differential equations, which reduce the time per training step for an NCDE by up to three orders of magnitude while achieving state-of-the-art performance across diverse time series benchmarks.

Modelisation of chaotic systems with a latent Stochastic Differential Equation

Stochastic Differential Equations (SDEs) have become a cornerstone of scientific machine learning, though they are predominantly utilized as algorithmic tools for uncertainty quantification or distribution matching. In contrast, leveraging SDEs fundamentally to model macroscopic, nonlinear physics as stochastic processes remains largely unexplored. This work introduces a probabilistic, non-intrusive reduced-order model (ROM) for chaotic dynamical systems. We argue that projecting high-dimensional nonlinear dynamics onto a low-dimensional manifold introduces irreducible uncertainty, compounded by the chaotic attractors and multi-admissible futures inherent to turbulent flows. Consequently, a chaotic system governed by a partial differential equation can be effectively modeled by an SDE in a suitable latent space. To this end, a nonlinear autoencoder is employed to map the flow field into a low-dimensional representation, within which the temporal evolution is explicitly governed by an SDE. The predictable component of the dynamics is captured by a learned drift term, while state-dependent stochasticity is absorbed by a diffusion term. We demonstrate that this probabilistic framework successfully propagates highly nonlinear states, offering a robust alternative to traditional deterministic methodologies for chaotic regimes. Ultimately, our model generates new chaotic flow trajectories that remain locally and globally consistent with the true transition kernel learned from Direct Numerical Simulation (DNS) data. Even though these generated trajectories are unique and distinct from the training set, they preserve the underlying statistics and manifolds, validating the strong generative performance and robustness of our methodology.

Accelerating transient probabilistic analysis of multidimensional nonlinear stochastic systems via sequential warm-start

Propensity Straight-Through Gradients for Discrete Stochastic Systems

Continuous-time Markov chains (CTMCs) provide the backbone for modeling discrete stochastic dynamics across applied, physical, and biological sciences. Their integration with modern gradient-based machine learning, however, is limited by the hard categorical event selection intrinsic to Gillespie-type simulation algorithms. We exploit the affine state update to obtain the exact one-step conditional-mean sensitivity by differentiating normalized reaction propensities. We pair this backward rule with exact forward trajectories to define the propensity straight-through (PST) estimator. At the trajectory level, we show that one-step sensitivities composed across events can depart from the exact multistep sensitivity. We derive the resulting per-step discrepancy in closed form and prove that it vanishes identically for affine downstream dependence. PST matches the accuracy of Gumbel-Softmax straight-through across all benchmarks: reversible dimerization (0.06% error), a genetic oscillator (1.7% error), a 50-task repressilator suite (0.17% median error), and patch-clamp ion-channel recordings ($R^2$ = 0.988). Under matched settings, PST converges 3.0-fold faster on the oscillator and 2.1-fold faster on the ion channel. At deep-learning scale, PST trains a 203,796-parameter stochastic reaction network with hard sampling, reaching 98.22% MNIST digit classification accuracy. By differentiating an exact conditional mean rather than a relaxed sample, PST offers a temperature- and Gumbel-free path to scalable gradient-based learning through exact stochastic trajectories.