Simultaneously Solving FBSDEs and their Associated Semilinear Elliptic PDEs with Small Neural Operators

Paper: arXiv 2410.14788

Abstract

Forward-backwards stochastic differential equations (FBSDEs) play an important role in optimal control, game theory, economics, mathematical finance, and in reinforcement learning. Unfortunately, the available FBSDE solvers operate on \textit{“individual”} FBSDEs, meaning that they cannot provide a computationally feasible strategy for solving large families of FBSDEs, as these solvers must be re-run several times. \textit{“Neural operators”} (NOs) offer an alternative approach for \textit{“simultaneously solving”} large families of decoupled FBSDEs by directly approximating the solution operator mapping \textit{“inputs:”} terminal conditions and dynamics of the backwards process to \textit{“outputs:”} solutions to the associated FBSDE. Though universal approximation theorems (UATs) guarantee the existence of such NOs, these NOs are unrealistically large. Upon making only a few simple theoretically-guided tweaks to the standard convolutional NO build, we confirm that ``small’’ NOs can uniformly approximate the solution operator to structured families of FBSDEs with random terminal time, uniformly on suitable compact sets determined by Sobolev norms using a logarithmic depth, a constant width, and a polynomial rank in the reciprocal approximation error. This result is rooted in our second result, and main contribution to the NOs for PDE literature, showing that our convolutional NOs of similar depth and width but grow only \textit{“quadratically”} (at a dimension-free rate) when uniformly approximating the solution operator of the associated class of semilinear Elliptic PDEs to these families of FBSDEs. A key insight into how NOs work we uncover is that the convolutional layers of our NO can approximately implement the fixed point iteration used to prove the existence of a unique solution to these semilinear Elliptic PDEs.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is dominated by advanced theoretical analysis of solution operators for FBSDEs and semilinear elliptic PDEs, relying heavily on Sobolev norms, approximation theory, and neural operator architecture tweaks; it contains no code, datasets, or backtesting, and the contributions are purely theoretical with no empirical validation.

Research Flowchart

  flowchart TD
  A["<b>Research Goal</b><br/>Simultaneously solve families of FBSDEs<br/>using small Neural Operators (NOs)"] --> B["<b>Methodology</b><br/>Build convolutional NO with<br/>theoretically-guided tweaks"]
  B --> C["<b>Inputs</b><br/>Terminal conditions & Dynamics<br/>of backward process"]
  C --> D["<b>Computational Process</b><br/>NO approximates solution operator<br/>via fixed-point iteration implementation"]
  D --> E["<b>Key Outcome 1</b><br/>Small NOs (log depth, const width)<br/>uniformly approximate FBSDE solutions"]
  D --> F["<b>Key Outcome 2</b><br/>NOs scale quadratically (dim-free)<br/>for semilinear Elliptic PDEs"]
  E --> G["<b>Impact</b><br/>Enables computationally feasible<br/>solutions for large families of FBSDEs"]
  F --> G

Cited References (180)

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7 of 180 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.