Inverse Problems with Machine Learning
Inverse problems infer hidden parameters or fields from indirect measurements, and machine learning provides fast, regularized estimators.
Forward versus inverse
A forward problem maps causes to effects: given material properties and sources, predict the observed signal. An inverse problem runs backward: given noisy measurements, recover the underlying parameters or fields that produced them. Inverse problems are often ill-posed, meaning the solution may be non-unique or extremely sensitive to measurement noise.
Classical framing
The standard approach minimizes a data-misfit term plus a regularizer that encodes prior knowledge, such as smoothness or sparsity. This Tikhonov-style formulation stabilizes the problem but requires many forward solves inside an optimization loop, which is expensive when each forward model is a full simulation.
How machine learning helps
- A learned surrogate replaces the costly forward model, so each optimization step is cheap
- A network can be trained to map measurements directly to parameters, giving instant estimates
- Physics-informed networks absorb unknown parameters as extra trainable variables
- Generative priors constrain solutions to a learned manifold of plausible fields
PINNs for inversion
A physics-informed network is naturally suited to inversion. Unknown coefficients in the governing equation, such as a diffusivity or a source strength, become trainable parameters alongside the network weights. Minimizing the combined residual and data loss recovers both the field and the unknown physics at once, without a separate outer loop.
Uncertainty is essential
Because inverse problems are ill-posed, a single best-fit answer can be misleading. Sound practice reports a distribution over solutions, using Bayesian methods, ensembles, or conformal bounds. In fusion, equilibrium reconstruction is a classic inverse problem: internal plasma profiles are inferred from external magnetic and other diagnostics, and quantifying the ambiguity in that reconstruction matters as much as the point estimate.