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Due to the complex behavior arising from non-uniqueness, symmetry, and bifurcations in the solution space, solving inverse problems of nonlinear differential equations (DEs) with multiple solutions is a challenging task. To address this, we propose homotopy physics-informed neural networks (HomPINNs), a novel framework that leverages homotopy continuation and neural networks (NNs) to solve inverse problems. The proposed framework begins with the use of NNs to simultaneously approximate unlabeled observations across diverse solutions while adhering to DE constraints. Through homotopy continuation, the proposed method solves the inverse problem by tracing the observations and identifying multiple solutions. The experiments involve testing the performance of the proposed method on one-dimensional DEs and applying it to solve a two-dimensional Gray-Scott simulation. Our findings demonstrate that the proposed method is scalable and adaptable, providing an effective solution for solving DEs with multiple solutions and unknown parameters. Moreover, it has significant potential for various applications in scientific computing, such as modeling complex systems and solving inverse problems in physics, chemistry, biology, etc. • Physics-informed neural networks are used to estimate unknown parameters with physical constraints. • HomPINNs helps to identify multiple solutions and solve inverse problems. • Numerical examples demonstrate its scalability and adaptability. • HomPINNs facilitates the discovery of previously unknown solutions. [ABSTRACT FROM AUTHOR]

Deep neural networks have been shown to provide accurate function approximations in high dimensions. However, fitting network parameters requires informative training data that are often challenging to collect in science and engineering applications. This work proposes Neural Galerkin schemes based on deep learning that generate training data with active learning for numerically solving high-dimensional partial differential equations. Neural Galerkin schemes build on the Dirac-Frenkel variational principle to train networks by minimizing the residual sequentially over time, which enables adaptively collecting new training data in a self-informed manner that is guided by the dynamics described by the partial differential equations. This is in contrast to other machine learning methods that aim to fit network parameters globally in time without taking into account training data acquisition. Our finding is that the active form of gathering training data of the proposed Neural Galerkin schemes is key for numerically realizing the expressive power of networks in high dimensions. Numerical experiments demonstrate that Neural Galerkin schemes have the potential to enable simulating phenomena and processes with many variables for which traditional and other deep-learning-based solvers fail, especially when features of the solutions evolve locally such as in high-dimensional wave propagation problems and interacting particle systems described by Fokker-Planck and kinetic equations. • Coupling training of deep networks with adaptive data acquisition to solve high-dimensional evolution equations. • Enables predicting local features in high dimensions for which training with classical, uniform data acquisition fails. • Sequentially collects data over time and so avoids requirement of having rich initial data set over the time-space domain. • Suited for transport problems, for which collocation-based deep-network methods with uniform data sampling are inefficient. [ABSTRACT FROM AUTHOR]

While deep learning algorithms demonstrate a great potential in scientific computing, its application to multi-scale problems remains to be a big challenge. This is manifested by the "frequency principle" that neural networks tend to learn low frequency components first. Novel architectures such as multi-scale deep neural network (MscaleDNN) were proposed to alleviate this problem to some extent. In this paper, we construct a subspace decomposition based DNN (dubbed SD2NN) architecture for a class of multi-scale problems by combining MscaleDNN algorithms with traditional numerical analysis ideas. The proposed architecture includes one low frequency normal DNN submodule, and one (or a few) high frequency MscaleDNN submodule(s), which are designed to capture the smooth part and the oscillatory part of the multi-scale solutions simultaneously. We demonstrate that the SD2NN outperforms existing models such as MscaleDNN, through several benchmark multi-scale problems in regular or perforated domains. • We propose SD2NN, a new architecture, combining subspace decomposition from numerical analysis and multiscale DNN solvers. • SD2NN has low/high frequency submodules, simultaneously capturing smooth/oscillatory parts of multi-scale solutions. • The new method outperforms state-of-the-art DNN solvers for 1D-3D multiscale PDEs on regular and perforated domains. [ABSTRACT FROM AUTHOR]