We address the problem of predicting the next states of a dynamical system governed by unknown temporal partial differential equations (PDEs) using only a short trajectory. While standard transformers provide a natural black-box solution to this task, the presence of a well-structured evolution operator in the data suggests a more tailored and efficient approach. Specifically, when the PDE is fully known, classical numerical solvers can evolve the state accurately with only a few parameters. Building on this observation, we introduce DISCO, a model that uses a large hypernetwork to process a short trajectory and generate the parameters of a much smaller operator network, which then predicts the next states through time integration. Our framework decouples dynamics estimation – i.e., DISCovering an evolution Operator from a short trajectory – from state prediction – i.e., evolving this operator. Experiments show that pretraining our model on diverse physics datasets achieves state-of-the-art performance while requiring significantly fewer epochs. Moreover, it generalizes well to unseen initial conditions and remains competitive when fine-tuned on downstream tasks.

Score-based diffusion models have significantly advanced high-dimensional data generation across various domains, by learning a denoising oracle (or score) from datasets. From a Bayesian perspective, they offer a realistic modeling of data priors and facilitate solving inverse problems through posterior sampling. Although many heuristic methods have been developed recently for this purpose, they lack the quantitative guarantees needed in many scientific applications. This work addresses the topic from two perspectives. We first present a hardness result indicating that a generic method leveraging the prior denoising oracle for posterior sampling becomes infeasible as soon as the measurement operator is mildly ill-conditioned. We next develop the *tilted transport* technique, which leverages the quadratic structure of the log-likelihood in linear inverse problems in combination with the prior denoising oracle to exactly transform the original posterior sampling problem into a new one that is provably easier to sample from. We quantify the conditions under which the boosted posterior is strongly log-concave, highlighting how task difficulty depends on the condition number of the measurement matrix and the signal-to-noise ratio. The resulting general scheme is shown to match the best-known sampling methods for Ising models, and is further validated on high-dimensional Gaussian mixture models.

Stochastic optimal control, which has the goal of driving the behavior of noisy systems, is broadly applicable in science, engineering and artificial intelligence. Our work introduces Stochastic Optimal Control Matching (SOCM), a novel Iterative Diffusion Optimization (IDO) technique for stochastic optimal control that stems from the same philosophy as the conditional score matching loss for diffusion models. That is, the control is learned via a least squares problem by trying to fit a matching vector field. The training loss, which is closely connected to the cross-entropy loss, is optimized with respect to both the control function and a family of reparameterization matrices which appear in the matching vector field. The optimization with respect to the reparameterization matrices aims at minimizing the variance of the matching vector field. Experimentally, our algorithm achieves lower error than all the existing IDO techniques for stochastic optimal control for three out of four control problems, in some cases by an order of magnitude. The key idea underlying SOCM is the path-wise reparameterization trick, a novel technique that may be of independent interest.

We perform a systematic study of the approximation properties and optimization dynamics of recurrent neural networks (RNNs) when applied to learn input-output relationships in temporal data. We consider the simple but representative setting of using continuous-time linear RNNs to learn from data generated by linear relationships. On the approximation side, we prove a direct and an inverse approximation theorem of linear functionals using RNNs, which reveal the intricate connections between memory structures in the target and the corresponding approximation efficiency. In particular, we show that temporal relationships can be effectively approximated by RNNs if and only if the former possesses sufficient memory decay. On the optimization front, we perform detailed analysis of the optimization dynamics, including a precise understanding of the difficulty that may arise in learning relationships with long-term memory. The term “curse of memory” is coined to describe the uncovered phenomena, akin to the “curse of dimension” that plagues high-dimensional function approximation. These results form a relatively complete picture of the interaction of memory and recurrent structures in the linear dynamical setting.

Recent years have witnessed the success of multi-agent reinforcement learning, which has motivated new research directions for mean-field control (MFC) and mean-field game (MFG), as the multi-agent system can be well approximated by a mean-field problem when the number of agents grows to be very large. In this paper, we study the policy gradient (PG) method for the linear-quadratic mean-field control and game, where we assume each agent has identical linear state transitions and quadratic cost functions. While most recent works on policy gradient for MFC and MFG are based on discrete-time models, we focus on a continuous-time model where some of our analyzing techniques could be valuable to the interested readers. For both the MFC and the MFG, we provide PG update and show that it converges to the optimal solution at a linear rate, which is verified by a synthetic simulation. For the MFG, we also provide sufficient conditions for the existence and uniqueness of the Nash equilibrium.

Machine learning models are changing the paradigm of molecular modeling, which is a fundamental tool for material science, chemistry, and computational biology. Of particular interest is the inter-atomic potential energy surface (PES). Here we develop Deep Potential - Smooth Edition (DeepPot-SE), an end-to-end machine learning-based PES model, which is able to efficiently represent the PES for a wide variety of systems with the accuracy of ab initio quantum mechanics models. By construction, DeepPot-SE is extensive and continuously differentiable, scales linearly with system size, and preserves all the natural symmetries of the system. Further, we show that DeepPot-SE describes finite and extended systems including organic molecules, metals, semiconductors, and insulators with high fidelity.