Approximation of Solution Operators for High-dimensional PDEs

TL;DR

The paper tackles the problem of efficiently approximating solution operators for high-dimensional evolution PDEs by a control-based paradigm. It represents PDE solutions with a reduced-order model u_θ and learns a vector field V_ξ in parameter space via Neural ODEs to steer θ along PDE-consistent trajectories, avoiding spatial discretization. A rigorous error analysis shows that, under mild assumptions, the method achieves provable approximation bounds for a broad class of second-order semilinear PDEs, with error growing at most like e^{Ct} and linearly in initial and projection errors. Numerically, the approach (i) surpasses prior operator-learning methods in accuracy and training efficiency, (ii) handles 10D heat and hyperbolic PDEs, and (iii) solves Hamilton-Jacobi-Bellman equations, producing accurate solutions with practical computation times. This framework thus provides a scalable, principled route to high-dimensional PDE solution operators, with potential for broad applications in control, finance, and physics-informed modeling.

Abstract

We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural network, we connect the evolution of the model parameters with trajectories in a corresponding function space. Using the computational technique of neural ordinary differential equation, we learn the control over the parameter space such that from any initial starting point, the controlled trajectories closely approximate the solutions to the PDE. Approximation accuracy is justified for a general class of second-order nonlinear PDEs. Numerical results are presented for several high-dimensional PDEs, including real-world applications to solving Hamilton-Jacobi-Bellman equations. These are demonstrated to show the accuracy and efficiency of the proposed method.

Figures (4)

• Figure 1: Comparison of NLS gaby2023neural method (black dashed line) and the proposed method (black solid line). While the previous method only allows accurate results for a small time scale $T=0.02$, the new method allows more accuracy for longer time scales $T=0.1$.
• Figure 2: Comparison with NLS gaby2023neural on the Heat Equation \ref{['eq:ivp-heat-eq']}. All plots show the $(x_1,x_2)$ plane. (First column) initial $g$; (Second column): reference solution; (Third column) Proposed method; and (Fourth column) NLS gaby2023neural. In all cases, the proposed method demonstrates significant improvement on solution accuracy.
• Figure 3: Mean relative error (dotted line) and standard deviation (grey) between the learned $u_{\theta(t)}$ and the reference solution $u^*$ for 100 different random initial conditions. (Left) hyperbolic PDE \ref{['eq:hyperbolic']}; (Right) Hamiltonian-Jacobi-Bellman equation \ref{['eq:ivp-hjb-eq']}. Note that in either case, the tested initials are not included in the training datasets.
• Figure 4: The evolution of 50 sampled points $X(0)$ (red circles) to time $X(1)$ (green triangles) in the (first column) $(x_1,x_2)$ plane; (second column) $(x_3,x_4)$ plane; (third column) $(x_5,x_6)$ plane; and (fourth column) $(x_7,x_8)$ plane. The background images show the expected minimum points of the terminal cost for the five randomly chosen initials. We see the solution $-\nabla u_{\theta(t)}$ provides correct control for all cases. Note that the control may not be able to steer very far initials (those started from the black regions) to the minimum (bright regions) since excessive movement causes large running costs that are not compensated by the terminal gains. As the terminal cost $g$ is scaled larger, this phenomenon becomes less likely to happen, and vice versa.

Theorems & Definitions (14)

• Definition 1: Sobolev ball
• Theorem 1
• Proposition 1
• Theorem 2
• proof
• Remark 1
• Lemma 1
• proof
• Remark 2
• Lemma 2: Theorem 5.1 deryck2021approximation