Inverse Problems
An inverse problem asks which inputs to a system (settings, compositions, model parameters) produce a desired set of outputs, the reverse of a forward problem in which outputs are predicted or measured from known inputs. A few models can be inverted analytically, but most are inverted numerically: one fits a forward model and searches over inputs with an optimizer, fits a backward model that maps outputs directly to inputs, or uses statistical approaches such as Bayesian inversion. Inverse problems can be ill-posed, with no solution for some outputs or several inputs giving the same output, so a method suited to one problem may fail on another.
Exhaustive enumeration for crystal design (2016)
Calfa and Kitchin posed materials design as the inverse of a kernel regression model whose inputs are a crystal’s space group and element counts, so any solution is directly a composition and structure (Calfa 2016). Because these inputs are discrete the problem is combinatorial, and as a first approach they enumerated every valid combination of predictor values present in the data, predicting properties for each and then filtering for targets such as a band gap near 1.23 eV or bounded Fermi levels. For 746 binary metal oxides this produced 1,153,504 candidates in about 19.7 h, including compositions absent from the training data; the authors note that model error can hide known crystals and present the cost as motivation for optimization-based formulations like those of computer-aided molecular design. Summary: calfa2016.
Inverse mapping by AD and the implicit function theorem (2023)
Alves, Kitchin and Lima computed inverse maps for process models without optimization (Alves et al. 2023). With a model written in JAX, automatic differentiation gives the Jacobian, the implicit function theorem converts it into derivatives of the inputs with respect to the outputs, and these are integrated with a fourth-order Runge-Kutta scheme along the discretized boundary of a target output region, starting from one known point, to trace the boundary of the input region that achieves it. For a CSTR design problem the traced boundary enclosed every brute-force grid point mapping into the target, and for a methane aromatization membrane reactor the result matched an IPOPT-based NLP inversion to within about 0.02% average relative difference while running in 0.93 min, compared with 17.84 min for a cold-start NLP with finite differences and 1.97 min for a warm-start NLP with AD derivatives. The input region found was non-convex even for a circular output target, and the authors flag singular Jacobians at bifurcations as a case needing arc-length continuation. This is the AD-based method later cited in the 2025 Comment below. Summary: alves2023.
Conventional and generative routes compared (2025)
Kitchin compared four ways to find the RGB LED settings on the Claude-Light instrument that give a target count in three light-sensor channels (Kitchin 2025). A linear forward model inverted with a minimizer and a partial least squares backward model gave the same answer as two generative models (a Gaussian mixture model and a ForestDiffusion flow-matching model) used by conditioning on the desired outputs and generating the inputs. He contrasts the trade-offs: the forward-plus-optimizer route needs an initial guess and can find multiple solutions from different guesses; the backward model is evaluated directly but must be built around the specific outputs to be fixed and fails for many-to-one mappings; and the generative models can condition on any subset of variables and give sample-based uncertainty. All of the data-driven methods returned confident but physically impossible or wrong inputs when asked for outputs outside the training data. The study covered only a linear, one-to-one case. Summary: kitchin2025a.
Automatic differentiation for inverse problems (2025)
Kitchin, Alves and Laird list inverse problems among the uses of differentiable programming outside ML training, citing an AD-based approach that maps two state spaces onto each other and that they describe as more efficient than conventional nonlinear programming formulations (Kitchin et al. 2025). The Comment gives no further detail on the method. Summary: kitchin2025.
Spectra-to-structure and generative design (2018)
Kitchin’s 2018 Comment on ML in catalysis points to two ML routes to inverse problems (Kitchin 2018). One inverts XANES measurements: thousands of candidate structures and their simulated spectra are generated, and a model is trained to map spectral features back to structure, an idea he notes could extend to any spectroscopy that can be simulated from a structural model. The other is generative design of molecules, in which a molecule is encoded in a continuous representation, optimized there and decoded back; he suggests it could help homogeneous catalysis and notes no analogue yet for solid-state materials, nanoparticles or process conditions. Summary: kitchin2018; see also Generative models.