Structure probing neural network deflation

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