We propose a new class of physics-informed neural networks, called Physics-Informed Generator-Encoder Adversarial Networks, to effectively address the challenges posed by forward, inverse, and mixed problems in stochastic differential equations. In these scenarios, while the governing equations are known, the available data consist of only a limited set of snapshots for system parameters. Our model consists of two key components: the generator and the encoder, both updated alternately by gradient descent. In contrast to previous approaches of directly matching the approximated solutions with real snapshots, we employ an indirect matching that operates within the lower-dimensional latent feature space. This method circumvents challenges associated with high-dimensional inputs and complex data distributions, while yielding more accurate solutions compared to existing neural network solvers. In addition, the approach also mitigates the training instability issues encountered in previous adversarial frameworks in an efficient manner. Numerical results provide compelling evidence of the effectiveness of the proposed method in solving different types of stochastic differential equations.

翻译：我们提出了一类新的物理信息神经网络——物理信息生成器-编码器对抗网络，旨在有效应对随机微分方程中正向、逆向和混合问题的挑战。在这些场景中，尽管控制方程已知，但可用数据仅为系统参数的有限快照集。该模型由两个关键组件构成：生成器和编码器，两者通过梯度下降交替更新。与以往直接将近似解与真实快照匹配的方法不同，我们采用了一种在低维潜在特征空间内进行的间接匹配策略。该方法规避了高维输入和复杂数据分布带来的挑战，同时相较于现有神经网络求解器，能够获得更精确的解。此外，该方式还高效缓解了先前对抗性框架中遇到的训练不稳定性问题。数值结果有力证明了所提方法在求解不同类型随机微分方程中的有效性。