Deep learning is a powerful tool for solving nonlinear differential equations, but usually, only the solution corresponding to the flattest local minimizer can be found due to the implicit regularization of stochastic gradient descent. This paper proposes a network-based structure probing deflation method to make deep learning capable of identifying multiple solutions that are ubiquitous and important in nonlinear physical models. First, we introduce deflation operators built with known solutions to make known solutions no longer local minimizers of the optimization energy landscape. Second, to facilitate the convergence to the desired local minimizer, a structure probing technique is proposed to obtain an initial guess close to the desired local minimizer. Together with neural network structures carefully designed in this paper, the new regularized optimization can converge to new solutions efficiently. Due to the mesh-free nature of deep learning, the proposed method is capable of solving high-dimensional problems on complicated domains with multiple solutions, while existing methods focus on merely one or two-dimensional regular domains and are more expensive in operation counts. Numerical experiments also demonstrate that the proposed method could find more solutions than exiting methods.

深度学习是求解非线性微分方程的强大工具，但是通常，由于随机梯度下降的隐式正则化，只能找到与最平坦的局部极小值相对应的解。本文提出了一种基于网络的结构探测放气方法，以使深度学习能够识别在非线性物理模型中普遍存在且很重要的多个解决方案。首先，我们引入使用已知解决方案构建的放气算子，以使已知解决方案不再是优化能量格局的局部最小化器。其次，为了促进向期望的局部最小化器的收敛，提出了一种结构探测技术来获得接近期望的局部最小化器的初始猜测。结合本文精心设计的神经网络结构，新的规范化优化可以有效地收敛到新的解决方案。由于深度学习的无网格性质，因此该方法能够通过多种解决方案在复杂域上解决高维问题，而现有方法仅关注一维或二维规则域，并且在操作数量上更为昂贵。数值实验还表明，所提出的方法比现有方法能找到更多的解决方案。