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Deep-MacroFin: Informed Equilibrium Neural Network for Continuous Time Economic Models

Yuntao Wu, Jiayuan Guo, Goutham Gopalakrishna, Zissis Poulos

TL;DR

Deep-MacroFin tackles high-dimensional PDEs arising in continuous-time economics by embedding economic structure into neural solvers that enforce HJB and market-clearing constraints. It combines MLPs and Kolmogorov-Arnold Networks with a time-stepping scheme to stabilize nonlinear HJB problems and enable efficient training. The framework demonstrates strong scalability and competitive performance against baselines, solving problems up to 100D in basic PDEs and up to 50D macro models, while delivering practical gains in memory and compute costs. It also reveals the symbolic interpretability of KANs for certain problems and identifies current challenges in scaling KANs, derivative caching overhead, and the need for automated loss balancing. Overall, Deep-MacroFin presents a versatile, economically informed approach to fast, high-dimensional equilibrium analysis in continuous-time finance and economics.

Abstract

In this paper, we present Deep-MacroFin, a comprehensive framework designed to solve partial differential equations, with a particular focus on models in continuous time economics. This framework leverages deep learning methodologies, including Multi-Layer Perceptrons and the newly developed Kolmogorov-Arnold Networks. It is optimized using economic information encapsulated by Hamilton-Jacobi-Bellman (HJB) equations and coupled algebraic equations. The application of neural networks holds the promise of accurately resolving high-dimensional problems with fewer computational demands and limitations compared to other numerical methods. This framework can be readily adapted for systems of partial differential equations in high dimensions. Importantly, it offers a more efficient (5$\times$ less CUDA memory and 40$\times$ fewer FLOPs in 100D problems) and user-friendly implementation than existing libraries. We also incorporate a time-stepping scheme to enhance training stability for nonlinear HJB equations, enabling the solution of 50D economic models.

Deep-MacroFin: Informed Equilibrium Neural Network for Continuous Time Economic Models

TL;DR

Deep-MacroFin tackles high-dimensional PDEs arising in continuous-time economics by embedding economic structure into neural solvers that enforce HJB and market-clearing constraints. It combines MLPs and Kolmogorov-Arnold Networks with a time-stepping scheme to stabilize nonlinear HJB problems and enable efficient training. The framework demonstrates strong scalability and competitive performance against baselines, solving problems up to 100D in basic PDEs and up to 50D macro models, while delivering practical gains in memory and compute costs. It also reveals the symbolic interpretability of KANs for certain problems and identifies current challenges in scaling KANs, derivative caching overhead, and the need for automated loss balancing. Overall, Deep-MacroFin presents a versatile, economically informed approach to fast, high-dimensional equilibrium analysis in continuous-time finance and economics.

Abstract

In this paper, we present Deep-MacroFin, a comprehensive framework designed to solve partial differential equations, with a particular focus on models in continuous time economics. This framework leverages deep learning methodologies, including Multi-Layer Perceptrons and the newly developed Kolmogorov-Arnold Networks. It is optimized using economic information encapsulated by Hamilton-Jacobi-Bellman (HJB) equations and coupled algebraic equations. The application of neural networks holds the promise of accurately resolving high-dimensional problems with fewer computational demands and limitations compared to other numerical methods. This framework can be readily adapted for systems of partial differential equations in high dimensions. Importantly, it offers a more efficient (5 less CUDA memory and 40 fewer FLOPs in 100D problems) and user-friendly implementation than existing libraries. We also incorporate a time-stepping scheme to enhance training stability for nonlinear HJB equations, enabling the solution of 50D economic models.
Paper Structure (45 sections, 73 equations, 13 figures, 17 tables, 2 algorithms)

This paper contains 45 sections, 73 equations, 13 figures, 17 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Methodology
  4. Problem formulation
  5. System design
  6. State variables and parameters
  7. Learnable variables
  8. Equations
  9. Conditions
  10. Endogenous equations
  11. HJB equations
  12. Constraints
  13. Systems
  14. Training
  15. Time-stepping scheme
  16. ...and 30 more sections

Figures (13)

  • Figure 1: System overview. This model assumes the presence of two state variables: $\eta$ (experts' wealth share) and $\sigma$ (volatility), two agents: $V_e$ (experts) and $V_h$ (households), and one endogenous variable: $q$ (price of the capital).
  • Figure 2: Laplace equation benchmark
  • Figure 3: Brunnermeier & Sannikov: Red shows the finite difference solutions by PyMacroFin; blue shows the fitted solution by Deep-MacroFin with MLP; orange shows the fitted solution by KAN.
  • Figure 4: Neoclassical growth model global solution
  • Figure 5: Lucas orchard benchmark
  • ...and 8 more figures