Source-linked AI summary
Solving high-dimensional partial differential equations using deep learning
Jiequn Han, Arnulf Jentzen, Weinan E
TL;DR
High-dimensional PDEs are hard to solve because of the curse of dimensionality, which makes computational cost grow exponentially with dimension. The paper reformulates nonlinear parabolic PDEs as BSDEs and approximates solution gradients with neural networks. Numerical examples report satisfactory accuracy and computational cost, while the method remains unable to handle the quantum many-body problem because of the Pauli exclusion principle.
Problem
The paper addresses the difficulty of practically solving high-dimensional PDEs when computational cost grows exponentially with dimension.
Method
The method reformulates nonlinear parabolic PDEs as BSDEs and approximates the gradient of the solution with deep neural networks.
Results
Numerical examples indicate satisfactory performance in both accuracy and computational cost, including a 0.17% relative error in the HJB example.
Takeaways & Limitations
The method opens possibilities for modeling multiple interacting agents, financial instruments, resources, or particles together rather than using ad hoc inter-relationship assumptions.
Takeaways & Limitations
The methodology cannot yet handle the quantum many-body problem because of the difficulty of dealing with the Pauli exclusion principle.
Abstract
from arXiv · showhide
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black-Scholes equation, the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up new possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their inter-relationships.
1 Introduction
High-dimensional PDEs are important in applications but are difficult to solve because computational cost grows exponentially with dimension. The paper applies deep neural networks to this challenge through a BSDE-based learning framework for nonlinear parabolic PDEs.
- High-dimensional PDEs arise in quantum many-body systems, financial derivatives, dynamic programming, games, and resource allocation.Examples include Schrödinger, nonlinear Black-Scholes, and Hamilton-Jacobi-Bellman equations.
- The curse of dimensionality makes practical use of high-dimensional PDE models difficult because solution cost increases exponentially with dimension.
- Traditional polynomial, piecewise-polynomial, wavelet, and basis-function approximations are likewise subject to the curse of dimensionality.
- Deep neural networks use compositional function representations and have shown remarkable practical success in artificial intelligence.
- The paper reformulates nonlinear parabolic PDEs as BSDEs and approximates solution gradients with neural networks.The approach is motivated by the analogy between the gradient and a policy function in deep reinforcement learning.
- The proposed framework is reported to perform satisfactorily in both accuracy and computational cost, extending deep learning to high-dimensional nonlinear PDEs.
Methodology
The methodology converts semilinear parabolic PDEs into a discretized BSDE computation and learns the gradient-related term with neural networks. Training minimizes terminal-condition mismatch using stochastic-gradient optimization.
- The method considers semilinear parabolic PDEs with specified coefficient functions, terminal condition, and target solution at an initial space-time point.
- The PDE solution is represented through a BSDE driven by a Brownian motion and a stochastic process.
- The initial solution and gradient are treated as parameters, while the BSDE is discretized over a time partition using an Euler scheme.
- At each time step, a multilayer feedforward neural network approximates x 7→σT(t, x) ∇u(t, x).
- The sub-networks are stacked into a deep network that maps sampled stochastic paths to an approximation of the terminal solution.
- The parameters are optimized with stochastic-gradient descent, using Adam in the numerical examples; the resulting methodology is called the deep BSDE method.
Examples
The deep BSDE method is tested on 100-dimensional nonlinear Black-Scholes, Hamilton-Jacobi-Bellman, and Allen-Cahn equations, achieving low reported relative errors across these examples. The Black-Scholes case also illustrates pricing with default risk, while the HJB case evaluates how optimal cost changes with control strength.
- Nonlinear Black-Scholes Equation with Default Risk: The nonlinear Black-Scholes example prices a European claim on 100 underlying assets while incorporating default risk.The model uses a piecewise-linear intensity function and a terminal condition based on the minimum asset value.
- Nonlinear Black-Scholes Equation with Default Risk: 0.46% relative error was obtained for the 100-dimensional nonlinear Black-Scholes equation.The approximation at t=0 and x=(100, ..., 100) used 40 time steps and required 1607 seconds.
- Nonlinear Black-Scholes Equation with Default Risk: Neglecting default risk changes the reference price from approximately 57.300 to approximately 60.781 in the Black-Scholes example.The paper reports that the deep BSDE method incorporates default risk into the pricing model.
- Hamilton-Jacobi-Bellman (HJB) Equation: The 100-dimensional HJB example achieves a 0.17% relative error in 330 seconds when λ = 1.The experiment evaluates the optimal cost at t=0 and x=(0, ..., 0).
- Hamilton-Jacobi-Bellman (HJB) Equation: In the HJB experiment, the optimal cost decreases as the control strength λ increases.The reported curve compares deep BSDE estimates across λ values with classical Monte Carlo simulations.
- Allen-Cahn Equation: The 100-dimensional Allen-Cahn example achieves a 0.30% relative error in 647 seconds.The experiment also computes the time evolution of the solution at x=(0, ..., 0) over t ∈ [0, 0.3].
Conclusions
The algorithm broadens high-dimensional PDE applications by enabling simultaneous treatment of many agents, instruments, or entities, while remaining unable to address the quantum many-body problem because of the Pauli exclusion principle.
- The algorithm enables economics models with many interacting agents instead of relying on a representative-agent model.
- In finance, the method can treat all participating instruments simultaneously rather than using ad hoc assumptions about their relationships.
- Operational-research problems with hundreds or thousands of participating entities can be handled directly without ad hoc approximations.
- The methodology cannot yet handle the quantum many-body problem because incorporating the Pauli exclusion principle is difficult.
Materials and Methods
The deep BSDE method converts nonlinear parabolic PDE solution into a forward stochastic computation, approximating spatial gradients with time-indexed neural subnetworks. These subnetworks are jointly optimized within a network whose accuracy improves with depth in a 100-dimensional test problem.
- BSDE reformulation: Markovian BSDEs connect nonlinear parabolic PDEs to processes satisfying Y_t = u(t, X_t) and Z_t = σ^T(t, X_t)∇u(t, X_t).
- BSDE reformulation: The method computes u(0, X_0) by substituting the BSDE identities, rewriting the equation forward, and temporally discretizing the resulting dynamics.
- Neural Network Architecture: At each time step, a multilayer feedforward neural network approximates the spatial gradient, with subnetworks connected by parameter-free forward iterations and state shortcuts.
- Neural Network Architecture: With H hidden layers per subnetwork and N time intervals, (H + 1)(N − 1) parameterized layers are optimized simultaneously.
- Neural Network Architecture: For solution values over a region D at t = 0, the method samples a non-degenerate D-valued initial state and adds networks for u(0, x) and ∇u(0, x).
- Implementation: The implementation uses fully connected ReLU subnetworks with batch normalization, TensorFlow, Adam optimization, batch size 64, and five independent runs for error statistics.
- Effect of Number of Hidden Layers: In a 100-dimensional reaction-diffusion test, increasing subnetwork depth from 0 to 4 hidden layers increases approximation accuracy.