Inverse Diffusivity

anypinn create my-project --template inverse-diffusivity

Recovers a space-dependent diffusivity \(D(x)\) represented as a neural network Field, rather than a single scalar parameter.

Background

Recovering spatially varying material properties from indirect measurements is a central problem in inverse theory, with applications in thermal conductivity imaging, permeability estimation in porous media, and non-destructive material testing. Unlike the other PDE templates where the unknown is a scalar, here the diffusivity \(D(x)\) is a function. The template represents it as a learnable neural network Field, demonstrating function-valued parameter recovery. The expanded form of the divergence operator \(\nabla \cdot (D\,\nabla u) = D\,u_{xx} + D'\,u_x\) shows how the spatial gradient of \(D\) couples to the solution gradient.

Governing Equations

\[ \frac{\partial u}{\partial t} = \nabla \cdot \bigl(D(x)\,\nabla u\bigr) = D(x)\,\frac{\partial^2 u}{\partial x^2} + D'(x)\,\frac{\partial u}{\partial x} \]

where:

• \(u(x, t)\): temperature / concentration field
• \(D(x)\): spatially varying diffusivity (to recover as a neural network Field)

Default Configuration

The generated template uses the following values.

Function to recover:

Symbol	Code constant	True profile
\(D(x)\)	TRUE_D_FN	\(0.1 + 0.05\,\sin(2\pi x)\)

The diffusivity ranges from \(0.05\) to \(0.15\) over the spatial domain.

Initial condition: \(u(x, 0) = \sin(\pi x)\)

Boundary conditions: \(u(0, t) = u(1, t) = 0\) (homogeneous Dirichlet).

Domain: \(x \in [0, 1], \quad t \in [0, 1]\)

Features Demonstrated

• Function-valued parameter recovery (\(D(x)\) as a Field)
• Composable differential operators from anypinn.lib.diff
• PDE inverse problem with spatially varying coefficients

Results