Multi-Trajectory Physics-Informed Neural Networks for HJB Equations with Hard Terminal Constraints: Optimal Execution and High-Dimensional LQR

Anthime Valin

Keywords: Partial Differential Equation (PDE), Optimal control, Hamilton-Jacobi-Bellman (HJB) equations, Physics-informed neural networks (PINNs), Multi-trajectory training, Linear-Quadratic Regulator (LQR), Hard-zero terminal state constraint, Optimal execution

TL;DR: We propose a multi-trajectory PINN to solve HJB equations with hard zero-terminal constraints, applying it to optimal trade execution and high-dimensional LQR, and achieving stronger terminal-state enforcement with reduced error optimal trajectories

Abstract: Hard terminal constraints are central to Hamilton-Jacobi-Bellman (HJB) formulations of optimal control, yet standard physics-informed neural networks (PINNs) trained with PDE residuals and soft terminal penalties often violate these constraints under rollout, yielding unstable controls near terminal. We propose Multi-Trajectory PINNs (MT-PINNs), which augment PINN training with a differentiable rollout-based loss that propagates terminal constraint penalties backward through time via backpropagation-through-time (BPTT). MT-PINNs directly couple PDE satisfaction with realized trajectory feasibility. We evaluate MT-PINNs on two representative HJB problems with hard terminal constraints. On a Gatheral-Schied optimal execution benchmark with hard-zero terminal inventory, MT-PINNs match closed-form solutions more closely along optimal paths and substantially tighten terminal-inventory enforcement relative to baseline PINNs. We further demonstrate the method on high-dimensional linear-quadratic regulation (LQR) HJBs, where MT-PINNs produce stable controls and strong terminal constraint satisfaction in dimensions where grid solvers are impractical.

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Journal Notes: Planned extensions include a systematic comparison of MT-PINNs with stronger baselines such as deep BSDE methods, neural operator, and recent neural HJB solvers. I will also provide clearer runtime and memory complexity analysis across methods.

Submission Number: 90