Heat Equation 1D

anypinn create my-project --template heat-1d

1D parabolic PDE inverse problem. Recovers thermal diffusivity \(\alpha\) from sparse temperature measurements.

Background

The heat equation describes thermal diffusion in a conducting medium according to Fourier's law. It is the prototypical parabolic PDE: an initial temperature profile smooths over time at a rate controlled by the thermal diffusivity \(\alpha\). The template uses a separable exact solution (exponential decay times a spatial mode) so that the recovered \(\alpha\) can be verified against the known analytical value. This makes it an ideal first example for PDE-based inverse problems.

Governing Equations

\[ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} \]

where:

• \(u(x, t)\): temperature field
• \(\alpha\): thermal diffusivity (to recover)

Exact solution used for data generation:

\[ u(x, t) = e^{-\alpha \pi^2 t}\,\sin(\pi x) \]

Default Configuration

The generated template uses the following values.

Parameters to recover:

Symbol	Code constant	True value
\(\alpha\)	TRUE_ALPHA	\(0.1\)

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

• PDE inverse problem (scalar parameter recovery)
• Sparse observation data
• Parabolic PDE handling

Results