The Dimension-Reduced Probability Density Evolution Equation (DR-PDEE) offers a promising approach for evaluating probability density evolution in stochastic dynamical systems. Physics-Informed Neural Networks (PINNs) are well-suited for solving DR-PDEE due to their ability to encode physical laws into the learning process. However, challenges arise from the spatio-temporal-dependence of unknown intrinsic drift and diffusion coefficients, which drive DR-PDEE, along with their derivatives. To address these challenges, a novel framework called Multi-Output Multi-Physics-Informed Neural Network (MO-MPINN) is proposed to predict the evolution of time-varying coefficients and response probability density simultaneously. MO-MPINN features multiple output neurons, eliminating the necessity for distinct identification of unknown spatio-temporal-dependent coefficients separately. It uses parallel subnetworks to reduce training complexity and embeds multiple physical laws in the loss function to ensure an accurate representation of the underlying principles. Leveraging automatic differentiation, MO-MPINN efficiently computes derivatives of coefficients without resorting to numerical differentiation. The framework is applicable to high-dimensional stochastic nonlinear systems with double randomness in structural parameters and excitations. Several structures are presented to validate the performance of the MO-MPINN. This study introduces a new paradigm for solving partial differential equations involving differentiation of spatio-temporal-dependent coefficients.

中文翻译：

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用于学习具有未知时空相关系数的降维概率密度演化方程的多输出多物理信息神经网络


降维概率密度演化方程（DR-PDEE）为评估随机动力系统中的概率密度演化提供了一种有前途的方法。物理信息神经网络 (PINN) 非常适合解决 DR-PDEE，因为它们能够将物理定律编码到学习过程中。然而，驱动 DR-PDEE 及其衍生物的未知固有漂移和扩散系数的时空依赖性带来了挑战。为了应对这些挑战，提出了一种称为多输出多物理信息神经网络（MO-MPINN）的新颖框架来同时预测时变系数和响应概率密度的演化。 MO-MPINN 具有多个输出神经元的特点，消除了单独识别未知时空相关系数的必要性。它使用并行子网络来降低训练复杂性，并在损失函数中嵌入多个物理定律，以确保准确表示基本原理。利用自动微分，MO-MPINN 可以有效计算系数的导数，而无需借助数值微分。该框架适用于结构参数和激励具有双重随机性的高维随机非线性系统。提出了几种结构来验证 MO-MPINN 的性能。这项研究引入了一种解决偏微分方程的新范式，涉及时空相关系数的微分。