Inverse Problems

Much of the physical world is inaccessible to direct measurement and can be reached only through the sparse traces it leaves in other observable variables. Here we introduce the Temporal Radiation Operator Network (TRON), a spatiotemporal neural operator that learns proxy-to-field mappings, reconstructing otherwise unobserved physical fields from sparse, indirect sensor histories. For cosmic-radiation mapping, TRON reconstructs sea-level effective dose at 65,341 locations from histories of 12 neutron monitors, achieving 0.096% mean relative $L_2$ error against EXPACS-generated reference fields in a chronologically held-out year; batched GPU inference evaluates a global field in approximately $\sim$1ms, compared with $\sim$57.6s for EXPACS on one CPU thread. On measured weather networks, TRON reconstructs temperature at held-out stations from thermometer or barometer histories, outperforming interpolation, calendar-based and hypsometric baselines. These results establish proxy-to-field reconstruction as a general route to recovering spatially distributed physical information beyond the direct reach of sparse sensor networks.

We study inverse problems where an unknown potential is observed only through samples from the measure it induces by a convex variational principle. Such problems arise in learning costs, energies, and dynamics from distributional data, but the associated forward solution map is typically nonlinear and implicit. We show that its optimality gap nevertheless yields convex empirical objectives for finite-dimensional potential classes, and we introduce sharpened Fenchel--Young losses that add a data-dependent discrepancy inside the forward problem. This keeps the estimator calibrated while improving the local geometry of the loss. Our main stability theorem separates the inverse error analysis into measurement error, forward perturbation, and empirical curvature. We instantiate this principle for inverse entropic unbalanced optimal transport and for inverse Jordan--Kinderlehrer--Otto (JKO) learning from independent snapshot samples, obtaining high-probability parameter recovery bounds. JKO schemes discretize Wasserstein gradient flows through a sequence of variational problems over measures, making them a natural language for population dynamics observed through snapshots. In this JKO case, the sharpened objective reduces to an unbalanced transport problem, which also clarifies the connection between variational gap losses and quadratic iJKO$^\star$ surrogates. Numerical experiments illustrate the conditioning effect of sharpening and its benefits for sparse inverse-gradient-flow recovery.

We introduce Superstate Quantum Mechanics (SQM), a theory that considers states in Hilbert space subject to multiple quadratic constraints, with energy'' also expressed as a quadratic function of these states. Traditional quantum mechanics corresponds to a single quadratic constraint of wavefunction normalization with energy expressed as a quadratic form involving the Hamiltonian. When SQM represents states as unitary operators, the stationary problem becomes a quantum inverse problem with multiple applications in physics, machine learning, and artificial intelligence. Any stationary SQM problem is equivalent to a new algebraic problem that we address in this paper. The non-stationary SQM problem considers the evolution of the system itself, involving the same energy'' operator as in the stationary case. Two possible options for the SQM dynamic equation are considered: (1) within the framework of linear maps from higher-order quantum theory, where 2D-type quantum circuits transform one quantum system into another; and (2) in the form of a Gross-Pitaevskii-type nonlinear map. Although no known physical process currently describes such 2D dynamics, this approach naturally bridges direct and inverse quantum mechanics problems, allowing for the development of a new type of computer algorithms. As an immediately available practical application of the theory, we consider using a quantum channel as a classical computational model; this type of computation can be performed on a classical computer.

Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with measurements. Consequently, these problems are naturally framed as stochastic inverse problems. In this paper, we explore three aspects of this problem: direct inversion, variational formulation with regularization, and optimization via gradient flows, drawing parallels with deterministic inverse problems. A key difference from the deterministic case is the space in which we operate. Here, we work within a probability space rather than Euclidean or Sobolev spaces, making tools from measure transport theory necessary for the study. Our findings reveal that the choice of metric---both in the design of the loss function and in the optimization process---significantly impacts the stability and properties of the optimizer.

Diffusion models excel in solving imaging inverse problems due to their ability to model complex image priors. However, their reliance on large, clean datasets for training limits their practical use where clean data is scarce. In this paper, we propose EMDiffusion, an expectation-maximization (EM) approach to train diffusion models from corrupted observations. Our method alternates between reconstructing clean images from corrupted data using a known diffusion model (E-step) and refining diffusion model weights based on these reconstructions (M-step). This iterative process leads the learned diffusion model to gradually converge to the true clean data distribution. We validate our method through extensive experiments on diverse computational imaging tasks, including random inpainting, denoising, and deblurring, achieving new state-of-the-art performance.

In the absence of conditioning information, normalizing flow models are trained using maximum likelihood. Previous extensions to conditional settings have abandoned the simple maximum-likelihood approach, as obtaining the full joint distribution of data and conditions for training purposes (or equivalently, the true conditional distribution for each point in the training dataset) is impossible. However, under special conditions that nevertheless apply to a wide class of useful problems, it is possible to obtain a simple, computationally cheap proxy with the same global minimum as the conditional likelihood. Conditional flow models trained using this proxy do not require explicit knowledge of the conditional distribution, auxiliary networks or other specific architecture, or additional loss terms beyond maximum likelihood. They provide a natural and powerful way to solve a class of ill-posed inverse problems. I demonstrate these properties on easily visualized toy problems, then use the method to successfully generate class-conditional images and to reconstruct semantically degraded images via super-resolution.