PINN Inverse Examples

PINN Inverse Examples#

This section contains examples demonstrating how to solve inverse problems using Physics-Informed Neural Networks (PINNs) with physical units. Inverse problems are among the most powerful applications of PINNs, allowing you to discover unknown parameters, fields, or even physical laws from observational data.

What are Inverse Problems?#

Inverse problems involve determining unknown parameters or functions within a governing equation using observed data. Unlike forward problems where everything is known except the solution, inverse problems require simultaneously:

Solving the PDE to find the solution field
Inferring unknown parameters such as:
- Material properties (diffusion coefficients, permeability, viscosity)
- Source terms or reaction rates
- Initial conditions or boundary values
- Spatially-varying fields (heterogeneous properties)

This is particularly valuable in real-world scenarios where:

Direct measurement of parameters is difficult or impossible
Only sparse observations of the solution are available
Physical properties vary in space or time
Multiple parameters need to be identified simultaneously

Why PINNs Excel at Inverse Problems#

Traditional inverse problem solvers often struggle with:

Ill-posedness: Multiple parameter sets may fit the data
Computational cost: Iterative optimization can be prohibitively expensive
Noise sensitivity: Measurement errors can lead to unstable solutions
High dimensionality: Many unknown parameters to identify

PINNs address these challenges by:

Embedding physical laws directly in the loss function (regularization)
Providing smooth, continuous representations of fields
Handling sparse and noisy data naturally
Enabling parameter discovery with minimal observations
Scaling to high-dimensional parameter spaces

Physical Units in Inverse Problems#

These examples use PINNx’s unit-aware framework, which is especially valuable for inverse problems:

Dimensional consistency: Ensures inferred parameters have correct physical units
Better conditioning: Unit normalization improves optimization stability
Physical constraints: Unit analysis helps constrain parameter search spaces
Interpretability: Results are directly usable without unit conversion
Validation: Easier to verify if discovered parameters are physically reasonable

Featured Examples#

The examples demonstrate parameter inference across multiple physics domains:

Diffusion and Transport#

1D Diffusion Inverse: Inferring diffusion coefficient from temperature measurements
Reaction-Diffusion: Discovering reaction rates in coupled systems

Fluid Dynamics#

Navier-Stokes Inverse: Identifying viscosity and pressure fields from velocity data
Brinkman-Forchheimer: Inferring permeability and form drag in porous media flow

Field Reconstruction#

Elliptic Inverse Field: Reconstructing spatially-varying coefficient fields
Heterogeneous Parameter Identification: Finding space-dependent material properties

Chemical Kinetics#

Reaction Inverse: Determining reaction rate constants from concentration data

Key Techniques Demonstrated#

Each example showcases important inverse problem methodologies:

Data assimilation: Incorporating sparse observations into the physics-informed loss
Multi-objective optimization: Balancing data fitting with PDE residuals
Uncertainty quantification: Assessing confidence in inferred parameters (where applicable)
Regularization strategies: Using physics to constrain ill-posed problems
Sparse data handling: Learning from limited measurements
Noise robustness: Dealing with measurement uncertainties
Parameter initialization: Effective starting points for optimization
Convergence monitoring: Tracking both solution and parameter convergence

Example Structure#

Each inverse problem example includes:

Problem formulation: Defining knowns and unknowns
Synthetic data generation: Creating observations (or using real data)
Network architecture: Designing networks for both solution and parameters
Loss function: Combining data misfit and PDE residuals
Training strategy: Optimization approaches for inverse problems
Results validation: Comparing inferred vs. true parameters
Sensitivity analysis: Understanding parameter identifiability

These examples serve as templates for solving your own inverse problems in physics and engineering.