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Laplace Transforms are used in circuit analysis. https://web.stanford.edu/~boyd/ee102/laplace_ckts.pdf

What do you do with it; check for stability, like a unit test?

Is it used in neural networks as approximators? From my understanding they're quite good but they suffer from overfitting.

For a brief period i did work on audio, video, and image processing along with signal processing.

Laplace and z-transform are widely used in control systems and signal processing. Both are more general forms of the Fourier transform in continuous and discrete time, respectively.