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Deep Learning (DL), in particular deep neural networks (DNN), by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, and hence promoting better generalization. This paper develops a model-constrained deep learning approach and its variant TNet—a Tikhonov neural network—that are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solution with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions in any desired accuracy for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for 1D deconvolution, 5 for inverse 2D heat conductivity problem, 100 for inverse initial conditions for time-dependent 2D Burgers’ equation, and 50 for inverse initial conditions for 2D Navier-Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. This is possible owing to the model-constrained term, replications, and randomization. 

Abstract This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems with Gaussian priors by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a sample average approximation (SAA). More importantly, we are able to prove a single theoretical result that guarantees the asymptotic convergence for a variety of randomized methods. Additionally, viewing randomized methods as an SAA enables us to prove, for the first time, a single non-asymptotic error result that holds for randomized methods under consideration. Another important consequence of our unified framework is that it allows us to discover new randomization methods. We present various numerical results for linear, nonlinear, algebraic, and PDE-constrained inverse problems that verify the theoretical convergence results and provide a discussion on the apparently different convergence rates and the behavior for various randomized methods.