Geometric Pareto Control: Physics-Supervised Pareto Representation Learning via Riemannian Energy-Gradient Flow

We study multi-objective sequential control problems in physical systems whose dynamics and operational constraints are known or can be represented by accurate physics-based models. Although reinforcement learning has been widely explored in such settings, standard policy learning still faces high-dimensional action search, fixed or externally supplied objective trade-offs, retraining under changed priorities, and hard feasibility requirements at deployment. Our premise is that known physics and constraints expose a compressible action structure that can be learned as a reusable response map before deployment. The resulting Pareto-relevant response family is organized before deployment, so online control begins with a physics-supervised prediction and local correction instead of repeatedly solving a nonlinear program. We propose Geometric Pareto Control (GPC), which embeds the supported family of dynamically feasible Pareto-optimal control responses into a continuous latent homotopy space using offline scalarized optimization under the known dynamics. At deployment, the measured physical state supplies priorities to the learned map; guarded local updates and domain-specific refinement adapt its decoded action or plan. We state formal assumptions under which decoded actions remain feasible and the action error does not accumulate over the rollout horizon. Across analytical control, safe multi-agent navigation, and optimal power flow, GPC compares favorably with optimization and safe-RL baselines in safety, feasibility, and real-time multi-objective performance. These results support physics-supervised learning of reusable Pareto-relevant responses that can be adapted online under changing priorities.