# Multi-Trajectory Physics-Informed Neural Networks for HJB Equations with Hard-Zero Terminal Inventory: Optimal Execution on Synthetic & SPY Data

arXiv: https://arxiv.org/abs/2512.12708 · HTML: https://mlsanity.com/paper/2512.12708/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 7 references that are in the map.
- First posted on arXiv 14 Dec 2025.

## Builds on (most cited first, 3 of 3)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [End-to-End Multi-Task Learning with Attention](https://mlsanity.com/paper/1803.10704/) (2018)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)

## Abstract (arXiv)

We study optimal trade execution with a hard-zero terminal inventory constraint, modeled via Hamilton-Jacobi-Bellman (HJB) equations. Vanilla PINNs often under-enforce this constraint and produce unstable controls. We propose a Multi-Trajectory PINN (MT-PINN) that adds a rollout-based trajectory loss and propagates a terminal penalty on terminal inventory via backpropagation-through-time, directly enforcing zero terminal inventory. A lightweight lambda-curriculum is adopted to stabilize training as the state expands from a risk-neutral reduced HJB to a risk-averse HJB. On the Gatheral-Schied single-asset model, MT-PINN aligns closely with their derived closed-form solutions and concentrates terminal inventory tightly around zero while reducing errors along optimal paths. We apply MT-PINNs on SPY intraday data, matching TWAP when risk-neutral, and achieving lower exposure and competitive costs, especially in falling windows, for higher risk-aversion.