ReLaTS: a Reinforcement Learning-based method for dynamically determining the coupling Time Step in multi-scale simulations of self-gravitating systems

Reinforcement learning in large or sparse-reward environments suffers from slow temporal-difference reward propagation, as value information spreads only locally across the state space. We propose Mesh-RL, a spatial domain-decomposition framework inspired by the finite element method and domain decomposition theory, which partitions the environment into overlapping subgrids and enforces boundary-consistent temporal-difference updates. Such an approach enables localized learning while ensuring globally coherent value propagation. Unlike hierarchical or model-based approaches, Mesh-RL accelerates long-range credit assignment without modifying the reward function, Bellman operator, or introducing explicit planning mechanisms. We evaluate Mesh-RL on hazard-dense grid-world environments with varying geometries and mesh resolutions. Across Q-learning, SARSA, and Dyna-Q, Mesh-RL consistently improves convergence speed, cumulative reward, and learning stability. Higher mesh resolutions sustain exploration, prevent premature convergence, and substantially accelerate value propagation to distant states. While Dyna-Q already benefits from internal planning, it still achieves additional gains under structured decomposition. Overall, Mesh-RL introduces a principled spatial domain-decomposition mechanism for accelerating temporal-difference learning. Our framework bridges finite element method-inspired boundary-consistency techniques from scientific computing with reinforcement learning to improve sample efficiency in sparse-reward environments. We will release source code of the study.

In science and engineering, Lagrangian simulation methods such as Smooth Particle Hydrodynamics (SPH) or Material Point Method (MPM) are often employed to study the behavior of dynamic systems. However, these methods can be prohibitively computationally expensive, particularly when simulating multi-scale spatial or temporal phenomena, e.g., void growth and coalescence within macro-scale geometries, structural failure of spacecraft components resulting from hypervelocity impact of space debris particles, etc. In contrast to graph-based methods, where the state of the system is understood as a discrete set of particles, we propose a learning framework for scalable representation and dynamics modeling of massive particle systems by treating the system state as a function and its evolution as a trajectory in Hilbert space. Rather than representing the state as a discrete set of particles or embedding it in a nonlinear latent manifold, we approximate the state space with a linear subspace spanned by learned neural basis functions. This parameterization enables direct projection to obtain latent coefficients and explicit access to the basis functions, avoiding optimization over a nonlinear latent space. The resulting representation admits a natural interpretation: latent variables correspond to coefficients in Hilbert space, and basis functions correspond to spatial modes, analogous to Proper Orthogonal Decomposition. The framework thus unifies classical projection-based reduced-order modeling with modern deep learning, while remaining invariant to the number of discretization points. Experiments on large-scale SPH simulations with over one million particles, including dynamic events with extreme deformation and fragmentation, demonstrate that the proposed method accurately reconstructs and predicts dynamics, achieving an R$^2$ score above $0.99$ with as few as $32$ basis functions.

Parallel trajectory optimization via the Alternating Direction Method of Multipliers (ADMM) has emerged as a scalable approach to long-horizon motion planning. However, existing frameworks typically decompose the problem into parallel subproblems based on a predefined fixed structure. Such structural rigidity often causes optimization stagnation in highly constrained regions, where a few lagging subproblems delay global convergence. A natural remedy is to adaptively re-split these stagnating segments online. Yet, deciding when, where, and how to split exceeds the capability of rule-based heuristics. To this end, we propose ATRS, a novel framework that embeds a shared Deep Reinforcement Learning policy into the parallel ADMM loop. We formulate this adaptive adjustment as a Multi-Agent Shared-Policy Markov Decision Process, where all trajectory segments act as homogeneous agents and share a unified neural policy network. This parameter-sharing architecture endows the system with size invariance, enabling it to handle dynamically changing segment counts during re-splitting and generalize to arbitrary trajectory lengths. Furthermore, our formulation inherently supports zero-shot generalization to unseen environments, as our network relies solely on the internal states of the numerical solver rather than on the geometric features of the environment. To ensure solver stability, a Confidence-Based Election mechanism selects only the most stagnating segment for re-splitting at each step. Extensive simulations demonstrate that ATRS accelerates convergence, reducing the number of iterations by up to 26.0% and the computation time by up to 19.1%. Real-world experiments further confirm its applicability to both large-scale offline global planning and real-time onboard replanning within 35 ms per cycle, with no sim-to-real degradation.