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This work presents a two-stage adaptive framework for progressivelydeveloping deep neural network (DNN) architectures that generalize well for agiven training data set. In the first stage, a layerwise training approach isadopted where a new layer is added each time and trained independently byfreezing parameters in the previous layers. We impose desirable structures onthe DNN by employing manifold regularization, sparsity regularization, andphysics-informed terms. We introduce a epsilon-delta stability-promotingconcept as a desirable property for a learning algorithm and show thatemploying manifold regularization yields a epsilon-delta stability-promotingalgorithm. Further, we also derive the necessary conditions for thetrainability of a newly added layer and investigate the training saturationproblem. In the second stage of the algorithm (post-processing), a sequence ofshallow networks is employed to extract information from the residual producedin the first stage, thereby improving the prediction accuracy. Numericalinvestigations on prototype regression and classification problems demonstratethat the proposed approach can outperform fully connected DNNs of the samesize. Moreover, by equipping the physics-informed neural network (PINN) withthe proposed adaptive architecture strategy to solve partial differentialequations, we numerically show that adaptive PINNs not only are superior tostandard PINNs but also produce interpretable hidden layers with provablestability. We also apply our architecture design strategy to solve inverseproblems governed by elliptic partial differential equations.

An Adaptive and Stability-Promoting Layerwise Training Approach for Sparse Deep Neural Network Architecture