Quantum-inspired nonlinear Galerkin ansatz for high-dimensional HJB equations

Paper: arXiv 2311.12239

Abstract

Neural networks are increasingly recognized as a powerful numerical solution technique for partial differential equations (PDEs) arising in diverse scientific computing domains, including quantum many-body physics. In the context of time-dependent PDEs, the dominant paradigm involves casting the approximate solution in terms of stochastic minimization of an objective function given by the norm of the PDE residual, viewed as a function of the neural network parameters. Recently, advancements have been made in the direction of an alternative approach which shares aspects of nonlinearly parametrized Galerkin methods and variational quantum Monte Carlo, especially for high-dimensional, time-dependent PDEs that extend beyond the usual scope of quantum physics. This paper is inspired by the potential of solving Hamilton-Jacobi-Bellman (HJB) PDEs using Neural Galerkin methods and commences the exploration of nonlinearly parametrized trial functions for which the evolution equations are analytically tractable. As a precursor to the Neural Galerkin scheme, we present trial functions with evolution equations that admit closed-form solutions, focusing on time-dependent HJB equations relevant to finance.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is highly mathematically dense, featuring advanced concepts like Galerkin methods, high-dimensional PDEs, and quantum mechanics analogies, but focuses on theoretical derivations and closed-form solutions for trial functions with no reported backtests or implementation metrics.

Research Flowchart

  flowchart TD
  Start["Research Goal: Solve high-dimensional HJB PDEs using<br>Neural Galerkin methods"] --> Method["Key Methodology:<br>Nonlinearly parametrized trial functions"]
  
  Method --> Inputs["Data/Inputs:<br>Time-dependent HJB equations<br>Finance applications"]
  
  Inputs --> Comp["Computational Process:<br>Analytically tractable evolution equations<br>Closed-form solutions"]
  
  Comp --> Outcomes["Key Findings:<br>1. Novel Neural Galerkin framework<br>2. Closed-form trial functions<br>3. Extension to high-dimensional PDEs"]

Cited References (12)

1. Normalizing flows for probabilistic modeling and inference (2021)
2. Solving the quantum many-body problem with artificial neural networks (2017)
3. DGM: A deep learning algorithm for solving partial differential equations (2018)
4. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations (2019) — link
5. Variational Monte Carlo approach to partial differential equations with neural networks (2022)
6. A deep learning algorithm for optimal investment strategies (2022)
7. An Example of Indifference Prices Under Exponential Preferences (2004) — link
8. A Valuation Algorithm for Indifference Prices in Incomplete Markets (2004) — link

0 of 12 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.