Additionally,though traditional PINNs (vanilla-PINNs) are typically stored andtrained in32-bit floating-point (fp32) ontheGPU, weshow that for DT-PINNs, using fp64 on the GPU leads to significantly faster training times than fp32 vanilla-PINNs with comparable accuracy. PINNscanbeusedboth to discover/infer PDEs that govern a given data set, and as direct PDE solvers.

Poisson equations underpin average-reward reinforcement learning, but beyond ergodicity they can be ill-posed, meaning that solutions are non-unique and standard fixed point iterations can oscillate on reducible or periodic chains. We study finite-state Markov chains with $n$ states and transition matrix $P$. We show that all non-decaying modes are captured by a real peripheral invariant subspace $\mathcal{K}(P)$, and that the induced operator on the quotient space $\mathbb{R}^n/\mathcal{K}(P)$ is strictly contractive, yielding a unique quotient solution. Building on this viewpoint, we develop an end-to-end pipeline that learns the chain structure, estimates an anchor based gauge map, and runs projected stochastic approximation to estimate a gauge-fixed representative together with an associated peripheral residual. We prove $\widetilde{O}(T^{-1/2})$ convergence up to projection estimation error, enabling stable Poisson equation learning for multichain and periodic regimes with applications to performance evaluation of average-reward reinforcement learning beyond ergodicity.

Physics-informed neural networks (PINNs) typically minimize average residuals, which can conceal large, localized errors. We propose Residual Risk-Aware Physics-Informed Neural Networks PINNs (RRaPINNs), a single-network framework that optimizes tail-focused objectives using Conditional Value-at-Risk (CVaR), we also introduced a Mean-Excess (ME) surrogate penalty to directly control worst-case PDE residuals. This casts PINN training as risk-sensitive optimization and links it to chance-constrained formulations. The method is effective and simple to implement. Across several partial differential equations (PDEs) such as Burgers, Heat, Korteweg-de-Vries, and Poisson (including a Poisson interface problem with a source jump at x=0.5) equations, RRaPINNs reduce tail residuals while maintaining or improving mean errors compared to vanilla PINNs, Residual-Based Attention and its variant using convolution weighting; the ME surrogate yields smoother optimization than a direct CVaR hinge. The chance constraint reliability level $α$ acts as a transparent knob trading bulk accuracy (lower $α$ ) for stricter tail control (higher $α$ ). We discuss the framework limitations, including memoryless sampling, global-only tail budgeting, and residual-centric risk, and outline remedies via persistent hard-point replay, local risk budgets, and multi-objective risk over BC/IC terms. RRaPINNs offer a practical path to reliability-aware scientific ML for both smooth and discontinuous PDEs.

Natural gradient methods significantly accelerate the training of Physics-Informed Neural Networks (PINNs), but are often prohibitively costly. We introduce a suite of techniques to improve the accuracy and efficiency of energy natural gradient descent (ENGD) for PINNs. First, we leverage the Woodbury formula to dramatically reduce the computational complexity of ENGD. Second, we adapt the Subsampled Projected-Increment Natural Gradient Descent algorithm from the variational Monte Carlo literature to accelerate the convergence. Third, we explore the use of randomized algorithms to further reduce the computational cost in the case of large batch sizes. We find that randomization accelerates progress in the early stages of training for low-dimensional problems, and we identify key barriers to attaining acceleration in other scenarios. Our numerical experiments demonstrate that our methods outperform previous approaches, achieving the same $L^2$ error as the original ENGD up to $75\times$ faster.

In this work, we present the first finite-time analysis of the Q-learning algorithm under time-varying learning policies (i.e., on-policy sampling) with minimal assumptions -- specifically, assuming only the existence of a policy that induces an irreducible Markov chain over the state space. We establish a last-iterate convergence rate for $\mathbb{E}[\|Q_k - Q^*\|_\infty^2]$, implying a sample complexity of order $O(1/ε^2)$ for achieving $\mathbb{E}[\|Q_k - Q^*\|_\infty] \le ε$, matching that of off-policy Q-learning but with a worse dependence on exploration-related parameters. We also derive an explicit rate for $\mathbb{E}[\|Q^{π_k} - Q^*\|_\infty^2]$, where $π_k$ is the learning policy at iteration $k$. These results reveal that on-policy Q-learning exhibits weaker exploration than its off-policy counterpart but enjoys an exploitation advantage, as its policy converges to an optimal one rather than remaining fixed. Numerical simulations corroborate our theory. Technically, the combination of time-varying learning policies (which induce rapidly time-inhomogeneous Markovian noise) and the minimal assumption on exploration presents significant analytical challenges. To address these challenges, we employ a refined approach that leverages the Poisson equation to decompose the Markovian noise corresponding to the lazy transition matrix into a martingale-difference term and residual terms. To control the residual terms under time inhomogeneity, we perform a sensitivity analysis of the Poisson equation solution with respect to both the Q-function estimate and the learning policy. These tools may further facilitate the analysis of general reinforcement learning algorithms with rapidly time-varying learning policies -- such as single-timescale actor--critic methods and learning-in-games algorithms -- and are of independent interest.

This paper introduces Deep Statistical Solvers (DSS), a new class of trainable solvers for optimization problems, arising e.g., from system simulations. The key idea is to learn a solver that generalizes to a given distribution of problem instances. This is achieved by directly using as loss the objective function of the problem, as opposed to most previous Machine Learning based approaches, which mimic the solutions attained by an existing solver.

We propose a continuous-time formulation of persistent contrastive divergence (PCD) for maximum likelihood estimation (MLE) of unnormalised densities. Our approach expresses PCD as a coupled, multiscale system of stochastic differential equations (SDEs), which perform optimisation of the parameter and sampling of the associated parametrised density, simultaneously. From this novel formulation, we are able to derive explicit bounds for the error between the PCD iterates and the MLE solution for the model parameter. This is made possible by deriving uniform-in-time (UiT) bounds for the difference in moments between the multiscale system and the averaged regime. An efficient implementation of the continuous-time scheme is introduced, leveraging a class of explicit, stable intregators, stochastic orthogonal Runge-Kutta Chebyshev (S-ROCK), for which we provide explicit error estimates in the long-time regime. This leads to a novel method for training energy-based models (EBMs) with explicit error guarantees.