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

ArXiv ID: 2311.12239 “View on arXiv”

Authors: Unknown

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.

Keywords: Neural Galerkin method, Hamilton-Jacobi-Bellman PDE, Partial differential equations, Nonlinearly parametrized trial functions, Stochastic optimization, Derivatives

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats
  • Why: 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.
  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"]