Learning Spatially-Varying Fractional Orders in PDEs

Hrishikesh Bhagwat, Pranav Janjani

Keywords: Fractional Order PDEs; Inverse Problems; Scientific Machine Learning

TL;DR: Learning spatially varying fractional orders in Fractional Order PDEs

Abstract:

Fractional differential equations generalize classical calculus
by incorporating non-local memory effects and anomalous diffusion,
capturing complex phenomena in viscoelastic materials, biological
tissue mechanics, and subsurface flow that integer-order models
struggle to represent.

Although fractional parameter estimation has been studied extensively
for spatially constant orders, comparatively little attention has
been paid to heterogeneous systems where $\alpha$ varies spatially.
However, such variation arises naturally in practice: a graded polymer
composite continuously transitions from a soft viscoelastic region
($\alpha \approx 0.3$, strong memory) to a stiff elastic region
($\alpha \approx 0.9$, near-classical behavior), a spatial pattern
that a single scalar $\alpha$ cannot truly represent.

We use the diffusive approximation to replace expensive fractional
derivative computations with auxiliary ODE systems, enabling
pointwise identification of $\alpha(x,y)$ through PDE residual
minimization. The estimated pointwise values are then interpolated
over the spatial domain using a neural network, without access to
ground truth fractional orders.

We evaluate our work in three benchmark cases of increasing
difficulty: smooth gradients, oscillatory patterns, and sharp
interfaces, achieving mean absolute errors below $0.05$ and
outperforming techniques by $4\times$. The diffusive approximation framework extends naturally to
a broad class of time-fractional PDEs beyond the Allen--Cahn
dynamics studied here.

This work contributes a systematic methodology for the identification of heterogeneous
fractional operators, with potential applications
in materials characterization, biophysics, and geophysical
modeling.

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Submission Number: 154