Late-Time Fractional-Order Identification in Caputo Diffusion Equation

What the math gives to ML

The paper supplies a mathematically explicit way to infer a fractional memory order from late-time decay using two scalar observations, together with the Caputo convolution that defines the underlying memory dynamics. The transferable asset is not the diffusion inverse problem itself, but the combination of power-law relaxation, resolvent-determined leading order, and a log-ratio estimator for the fractional exponent. A practical neural-network adaptation is an optimizer whose memory exponent is selected online from the late-time decay of a smoothed loss or gradient norm, then used to control a fractional-memory update. The diffusion assumptions do not hold exactly for nonconvex neural training, so the method should be treated as an adaptive optimizer heuristic and tested against AdamW and fixed-order fractional optimizers.