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   We propose a deep learning solution to the inverse problem of localizing sources of network diffusion. Invoking graph signal processing (GSP) fundamentals, the problem boils down to blind estimation of a diffusion filter and its sparse input signal encoding the source locations. While the observations are bilinear functions of the unknowns, a mild requirement on invertibility of the graph filter enables a convex reformulation that we solve via the alternating-direction method of multipliers (ADMM). We unroll and truncate the novel ADMM iterations, to arrive at a parameterized neural network architecture for Source Localization on Graphs (SLoG-Net), that we train in an end-to-end fashion using labeled data. This way we leverage inductive biases of a GSP model-based solution in a data-driven trainable parametric architecture, which is interpretable, parameter efficient, and offers controllable complexity during inference. Experiments with simulated data corroborate that SLoG-Net exhibits performance in par with the iterative ADMM baseline, while attaining significant (post-training) speedups. 

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   Unlike traditional structural materials, soft solids can often sustain very large deformation before failure, and many exhibit nonlinear viscoelastic behavior. Modeling nonlinear viscoelasticity is a challenging problem for a number of reasons. In particular, a large number of material parameters are needed to capture material response and validation of models can be hindered by limited amounts of experimental data available. We have developed a Gaussian Process (GP) approach to determine the material parameters of a constitutive model describing the mechanical behavior of a soft, viscoelastic PVA hydrogel. A large number of stress histories generated by the constitutive model constitute the training sets. The low-rank representations of stress histories by Singular Value Decomposition (SVD) are taken to be random variables which can be modeled via Gaussian Processes with respect to the material parameters of the constitutive model. We obtain optimal material parameters by minimizing an objective function over the input set. We find that there are many good sets of parameters. Further the process reveals relationships between the model parameters. Results so far show that GP has great potential in fitting constitutive models. 

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5. Learning to identify sources of network diffusion

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