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Developments in the inverse problem for extreme-mass-ratio inspirals
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Solving Forward and Inverse Problems for Hyperspectral Satellite Remote Sensors using Principal Component Analysis
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Leveraging a Machine Learning Approach to Solve an Inverse Problem Performance Query.
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Hyper-differential sensitivity analysis of PDE-constrained inverse problems.
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Solving Complex Inverse Problems for Design and Manufacturing Using High-Performance Computing.
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Solving inverse problems for process-structure linkages using asynchronous parallel Bayesian optimization.
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Control volume PINNs: a method for solving inverse problems with hyperbolic PDEs.
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Enabling and interpreting hyper-differential sensitivity analysis for Bayesian inverse problems.
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Inverse Problem Approach to Spacecraft Charging Simulations
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Hyper-differential sensitivity analysis in PDE-constrained inverse problems.
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A Spatially Varying Bayesian Approach to Solving Inverse Problems: Applications to Deblurring Cygnus Radiographs
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Particle approach to inverse problems via gradient-based optimization.
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On the inversion of eigenvalue problems
Inverse scattering problem in quantum mechanics - inversion of eigenvalue problems
Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion
Partial differential equation (PDE)-governed inverse problems are fundamental across various scientific and engineering applications; yet they face significant challenges due to nonlinearity, ill-posedness, and sensitivity to noise. Here, we introduce a computational framework, regularization by denoising using diffusion models for partial differential equations (RED-DiffEq), by integrating physics-driven inversion and data-driven learning. RED-DiffEq leverages pretrained diffusion models as a regularization mechanism for PDE-governed inverse problems. We apply RED-DiffEq to solve the full waveform inversion problem in geophysics, a challenging seismic imaging technique that seeks to reconstruct high-resolution subsurface velocity models from seismic measurement data. Our method shows enhanced accuracy and robustness compared to benchmark methods. Additionally, it exhibits strong generalization and domain decomposition capacity, enabling the inversion of more complex velocity models with larger domains than those used in training the diffusion model. Our framework can also be directly applied to diverse PDE-governed inverse problems.
Input specific neural networks
Neural networks have emerged as powerful tools for mapping between inputs and outputs. However, their black-box nature limits the ability to encode or impose specific structural relationships between inputs and outputs. Many scientific and engineering problems, such as constitutive modeling in solid mechanics, require networks that can enforce convexity, monotonicity, or other structural constraints to ensure physical consistency. Here, we introduce the Input Specific Neural Network (ISNN), a new architecture that enables multiple, distinct constraints to be imposed on different input subsets for scalar-valued outputs. This framework unifies convex, monotone–convex, monotone, and arbitrary mappings within a single network for the first time. Two ISNN architectures with analytical first- and second-order derivatives are developed. We demonstrate the performance on synthetic toy problems, inverse problems in isotropic hyperelasticity, and finite element simulations. ISNNs achieve improved extrapolation behavior, require fewer invariant inputs than standard input convex networks for polyconvex potentials, and enable significant computational savings via manual differentiation. We also show how ISNNs can be used to learn structural relationships between inputs and outputs via a binary gating mechanism. Particularly, ISNNs are employed to model a homogenized anisotropic free energy potential in a decoupled multiscale setting. The network learns whether or not the potential should be modeled as polyconvex and retains only the relevant layers while using the minimum number of inputs. ISNNs provide a flexible foundation for embedding structural priors into neural networks, enhancing both interpretability and stability. They are broadly applicable across computational mechanics and other scientific domains requiring constrained functional relationships.
The inverse scattering problem at fixed angular momentum for nonlocal separable interactions
The problem of inverse scattering at fixed angular momentum is considered. The problem is particularized to the case of nonlocal separable interactions. A brief survey of the inverse problem for nonlocal separable interactions is presented. This problem can be solved exactly by integration. It amounts to solving singular integral equations of the Hilbert-Mushkhelishvili type, which have been studied extensively in the past and appear in many areas of physics, including theory of elasticity and dispersions relations in high energy physics.