Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs
This work addresses computational bottlenecks in PINNs for high-dimensional PDEs, offering practical solutions for researchers in scientific computing and machine learning, though it is incremental by building on existing PINN methods.
The paper tackles the computational cost and bias issues in physics-informed neural networks (PINNs) for high-dimensional PDEs by introducing Randomized Smoothing PINN (RS-PINN) with bias correction and a hybrid method, achieving improved efficiency and accuracy across diverse PDEs like Fokker-Planck and Burgers' equations.
While physics-informed neural networks (PINNs) have been proven effective for low-dimensional partial differential equations (PDEs), the computational cost remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling Monte Carlo methods for derivative approximation, eliminating the need for costly auto-differentiation. Despite its computational efficiency in high dimensions, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD). We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss and the PDE nonlinearity. We propose tailored bias correction techniques based on the order of PDE nonlinearity. The unbiased RS-PINN allows for a detailed examination of its pros and cons compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, HJB, viscous Burgers', Allen-Cahn, and Sine-Gordon equations, illustrate the bias-variance trade-off and highlight the effectiveness of the hybrid RS-PINN. Empirical guidelines are provided for selecting biased, unbiased, or hybrid versions, depending on the dimensionality and nonlinearity of the specific PDE problem.