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.

翻译：尽管物理信息神经网络（PINNs）已证明在低维偏微分方程（PDEs）中行之有效，但其计算成本仍是高维场景面临的障碍。当在物理信息损失函数中计算高阶高维导数时，这一问题尤为突出。随机平滑PINN（RS-PINN）通过引入高斯噪声对原始神经网络模型进行随机平滑，利用蒙特卡洛方法近似导数，从而无需昂贵的自动微分。尽管该方法在高维计算中具有效率优势，但RS-PINN在损失函数和梯度中均引入偏差，对收敛性产生负面影响，尤其是与随机梯度下降（SGD）结合时。我们系统分析了RS-PINN的偏差来源，将其归因于均方误差（MSE）损失的非线性与PDE的非线性。针对PDE非线性阶数，我们提出了定制化的偏差校正技术。无偏RS-PINN使我们能够详细比较其与有偏版本的优缺点：具体而言，有偏版本方差更低、运行速度更快，但受偏差影响精度较差。为优化偏差-方差权衡，我们通过混合方法结合两种方案，平衡有偏版本的快速收敛性与无偏版本的高精度。此外，我们提出了RS-PINN的增强实现。通过在多种高维PDE（包括福克-普朗克方程、HJB方程、粘性伯格斯方程、艾伦-卡恩方程和正弦-戈尔登方程）上的大量实验，验证了偏差-方差权衡现象，并凸显了混合RS-PINN的有效性。最后，我们提供了选择有偏、无偏或混合版本的经验准则，具体取决于特定PDE问题的维度与非线性程度。