Inverse Problems

underPINN supports joint optimisation of network weights and physics parameters, recovering unknown PDE coefficients directly from sparse, noisy observations.

examples/heat/inverse.py recovers the unknown thermal diffusivity α from 50 sparse noisy observations of a 1-D diffusion field.

How it works

Mechanism	Description
Joint optimisation	The optimizer simultaneously updates network weights θ and the physics parameter log_α = log(α) via a single jax.grad call.
Log-parameterisation	Optimising log_α instead of α directly guarantees positivity without any constraints or projections; the true α is recovered as exp(log_α) after training.
Observation loss	A separate MSE term penalises the discrepancy between model predictions at the 50 observation locations and the noisy measurements; the PDE residual loss acts as the regulariser.

# Simplified view of the inverse problem setup
from underPINN.pde.diffusion import DiffusionInversePDE

pde = DiffusionInversePDE(model, log_alpha_init=jnp.log(0.5))
# pde.log_alpha is a trainable parameter alongside model weights
# After training: alpha_recovered = jnp.exp(pde.log_alpha)

The 2-D diffusion inverse case (examples/inverse/inverse_diffusion.py) follows the same pattern for a full 2-D domain.

Inverse-problem examples

Example	Recovers	Config
1-D Heat — Inverse	Thermal diffusivity α from 50 noisy observations	examples/heat/heat_inverse.yaml
2-D Diffusion Inverse	α via log-parameterised joint optimisation	examples/inverse/config.yaml

Gradient flow

Because log_α is just another leaf in the same parameter pytree as the network weights, gradients flow through both the PDE residual and the observation loss simultaneously — no alternating-optimisation scheme or bi-level loop is required.

See also

PDE Library Reference for the full list of PDE residual classes, including DiffusionInversePDE.