Finding the inverse of a matrix with neural networks
In general this problem falls under the umbrella of "structured prediction" since you are trying to estimate a number of things that are related by virtue of being embedded in a PSD matrix.

Instead of estimating the inverse in one swoop, I'd pick an appropriate algorithm, say inversion by eigendecomposition, estimate the components (eigenvalues and eigenvectors), then piece them together. Here's one paper that shows how: Neural networks based approach for computing eigenvectors and eigenvalues of symmetric matrix.

Alternatively, you can investigate approximation algorithms such as the one elaborated in Approximating the inverse of a symmetric positive definite matrix

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