Inverse Problem
Definition
Inferring model parameters, source terms or boundary conditions from observations — the opposite of the forward problem of computing results from known parameters. Usually ill-posed (non-unique or unstable), yet among the most valuable problems in engineering.
What it is
Forward problem: given parameters p, compute the field u. Inverse problem: given partial observations u_obs, recover p. Inverse problems are typically ill-posed in the Hadamard sense: the solution may be non-unique (different parameters produce identical observations) or may depend discontinuously on noisy data.
Key points
- Regularization is mandatory: Tikhonov, sparsity priors, physical constraints, Bayesian priors — all inject extra information to make the problem well-posed.
- Two values of AI: (a) surrogates make the expensive forward problem cheap enough for optimization or sampling; (b) differentiable solvers give exact gradients instead of finite differences.
- PINNs fit naturally: the solution is analytic and differentiable, so parameters and solution can be optimized jointly and inversion code stays short.
- Uncertainty is required: report a posterior distribution or confidence interval; a point estimate is often misleading.
Common pitfalls / boundaries
- A good fit does not mean the correct parameters — non-uniqueness means another parameter set may fit equally well.
- Observation noise, sensor placement and regularization weight jointly determine the result; sensitivity analysis is not optional.