# A Schr"odinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control

arXiv: https://arxiv.org/abs/2603.23173 · HTML: https://mlsanity.com/paper/2603.23173/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 76 references that are in the map.
- First posted on arXiv 24 Mar 2026.

## Builds on (most cited first, 27 of 27)

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- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space](https://mlsanity.com/paper/2005.05409/) (2020)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
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- [The SSL Interplay: Augmentations, Inductive Bias, and Generalization](https://mlsanity.com/paper/2302.02774/) (2023)
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- [Stochastic Optimal Control for Collective Variable Free Sampling of Molecular Transition Paths](https://mlsanity.com/paper/2207.02149/) (2022)
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- [Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks](https://mlsanity.com/paper/2110.14523/) (2021)
- [A Taxonomy of Loss Functions for Stochastic Optimal Control](https://mlsanity.com/paper/2410.00345/) (2024)
- [Convergence of a robust deep FBSDE method for stochastic control](https://mlsanity.com/paper/2201.06854/) (2022)
- [Solving stochastic optimal control problem via stochastic maximum principle with deep learning method](https://mlsanity.com/paper/2007.02227/) (2020)
- [The Galerkin method beats Graph-Based Approaches for Spectral Algorithms](https://mlsanity.com/paper/2306.00742/) (2023)
- [Mean field control hierarchy](https://mlsanity.com/paper/1608.01728/) (2016)
- [Schr"odinger Approach to Optimal Control of Large-Size Populations](https://mlsanity.com/paper/1810.06064/) (2018)
- [A mean-field game model for homogeneous flocking](https://mlsanity.com/paper/1803.05250/) (2018)
- [Error Analysis of Option Pricing via Deep PDE Solvers: Empirical Study](https://mlsanity.com/paper/2311.07231/) (2023)
- [On a Mean Field Optimal Control Problem](https://mlsanity.com/paper/1909.10596/) (2019)
- [Space mapping-based Receding Horizon Control for Stochastic Interacting Particle Systems: dogs herding sheep](https://mlsanity.com/paper/1910.05719/) (2019)
- [A Spectral Approach to Optimal Control of the Fokker-Planck Equation](https://mlsanity.com/paper/2503.15125/) (2025)

## Abstract (arXiv)

High-dimensional stochastic optimal control (SOC) becomes harder with longer planning horizons: existing methods scale linearly in the horizon T, with performance often deteriorating exponentially. We overcome these limitations for a subclass of linearly-solvable SOC problems-those whose uncontrolled drift is the gradient of a potential. In this setting, the Hamilton-Jacobi-Bellman equation reduces to a linear PDE governed by an operator L. We prove that, under the gradient drift assumption, L is unitarily equivalent to a Schr"odinger operator S = -Delta + V with purely discrete spectrum, allowing the long-horizon control to be efficiently described via the eigensystem of L. This connection provides two key results: first, for a symmetric linear-quadratic regulator (LQR), S matches the Hamiltonian of a quantum harmonic oscillator, whose closed-form eigensystem yields an analytic solution to the symmetric LQR with arbitrary terminal cost. Second, in a more general setting, we learn the eigensystem of L using neural networks. We identify implicit reweighting issues with existing eigenfunction learning losses that degrade performance in control tasks, and propose a novel loss function to mitigate this. We evaluate our method on several long-horizon benchmarks, achieving an order-of-magnitude improvement in control accuracy compared to state-of-the-art methods, while reducing memory usage and runtime complexity from O(Td) to O(d).