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Solving Heterogeneous Agent Models with Physics-informed Neural Networks

Marta Grzeskiewicz

TL;DR

This paper tackles the computational challenges of continuous-time heterogeneous agent macro models by introducing ABH-PINN, a physics-informed neural network solver that embeds the Hamilton-Jacobi-Bellman and Kolmogorov Forward equations directly into neural networks. By replacing grid-based discretisation with mesh-free learning, the ABH-PINN framework aims to mitigate the curse of dimensionality, improve solution smoothness, and enable faster policy analysis. The authors provide a detailed methodology for training two PINNs that approximate the value function $v(a,z,t)$ and the distribution $g(a,z,t)$, along with procedures to update aggregate capital $K$ and macro-prices $r$ and $w$, and demonstrate preliminary results matching finite-difference solvers in a canonical ABH setup. The work highlights significant potential for extending to higher state dimensions, incorporating data, and applying PINN-based solvers to a broader class of economic PDE systems, thereby advancing computational macroeconomics and probabilistic policy evaluation.

Abstract

Understanding household behaviour is essential for modelling macroeconomic dynamics and designing effective policy. While heterogeneous agent models offer a more realistic alternative to representative agent frameworks, their implementation poses significant computational challenges, particularly in continuous time. The Aiyagari-Bewley-Huggett (ABH) framework, recast as a system of partial differential equations, typically relies on grid-based solvers that suffer from the curse of dimensionality, high computational cost, and numerical inaccuracies. This paper introduces the ABH-PINN solver, an approach based on Physics-Informed Neural Networks (PINNs), which embeds the Hamilton-Jacobi-Bellman and Kolmogorov Forward equations directly into the neural network training objective. By replacing grid-based approximation with mesh-free, differentiable function learning, the ABH-PINN solver benefits from the advantages of PINNs of improved scalability, smoother solutions, and computational efficiency. Preliminary results show that the PINN-based approach is able to obtain economically valid results matching the established finite-difference solvers.

Solving Heterogeneous Agent Models with Physics-informed Neural Networks

TL;DR

This paper tackles the computational challenges of continuous-time heterogeneous agent macro models by introducing ABH-PINN, a physics-informed neural network solver that embeds the Hamilton-Jacobi-Bellman and Kolmogorov Forward equations directly into neural networks. By replacing grid-based discretisation with mesh-free learning, the ABH-PINN framework aims to mitigate the curse of dimensionality, improve solution smoothness, and enable faster policy analysis. The authors provide a detailed methodology for training two PINNs that approximate the value function and the distribution , along with procedures to update aggregate capital and macro-prices and , and demonstrate preliminary results matching finite-difference solvers in a canonical ABH setup. The work highlights significant potential for extending to higher state dimensions, incorporating data, and applying PINN-based solvers to a broader class of economic PDE systems, thereby advancing computational macroeconomics and probabilistic policy evaluation.

Abstract

Understanding household behaviour is essential for modelling macroeconomic dynamics and designing effective policy. While heterogeneous agent models offer a more realistic alternative to representative agent frameworks, their implementation poses significant computational challenges, particularly in continuous time. The Aiyagari-Bewley-Huggett (ABH) framework, recast as a system of partial differential equations, typically relies on grid-based solvers that suffer from the curse of dimensionality, high computational cost, and numerical inaccuracies. This paper introduces the ABH-PINN solver, an approach based on Physics-Informed Neural Networks (PINNs), which embeds the Hamilton-Jacobi-Bellman and Kolmogorov Forward equations directly into the neural network training objective. By replacing grid-based approximation with mesh-free, differentiable function learning, the ABH-PINN solver benefits from the advantages of PINNs of improved scalability, smoother solutions, and computational efficiency. Preliminary results show that the PINN-based approach is able to obtain economically valid results matching the established finite-difference solvers.

Paper Structure

This paper contains 31 sections, 21 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Relevant Literature
  3. PINNs in Economics and Finance
  4. Heterogeneous Agent Models in Macroeconomics
  5. Background
  6. ABH Economy
  7. Setup
  8. Individuals.
  9. Firms.
  10. Recursive formulation
  11. HJB equation.
  12. KF equation.
  13. PINNs
  14. Methodology
  15. General Framework
  16. ...and 16 more sections

Figures (5)

  • Figure 1: Value, consumption and density functions as trained by ABH-PINN at $t=1$.
  • Figure 2: Value, consumption and density functions as trained by ABH-PINN at $t=2$.
  • Figure 3: Value, consumption and density functions as trained by ABH-PINN at $t=5$.
  • Figure 4: Value, consumption and density functions as trained by ABH-PINN at $t=9$.
  • Figure 5: Computed time paths.