Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks

Understand

Physics-Informed Neural Networks (PINNs) have proven effective in solving partial differential equations (PDEs), especially when some data are available by seamlessly blending data and physics.

• However, extending PINNs to high-dimensional and even high-order PDEs encounters significant challenges due to the computational cost associated with automatic differentiation in the residual loss.
• Herein, we address the limitations of PINNs in handling high-dimensional and high-order PDEs by introducing Hutchinson Trace Estimation (HTE).
• Starting with the second-order high-dimensional PDEs ubiquitous in scientific computing, HTE transforms the calculation of the entire Hessian matrix into a Hessian vector product (HVP).