Report on Current Developments in Inverse Problem Solvers for PDE-Constrained Systems
General Direction of the Field
The field of inverse problem solvers for partial differential equation (PDE)-constrained systems is witnessing a significant shift towards more sophisticated and robust methodologies. Recent advancements are characterized by the integration of deep learning techniques with traditional optimization methods, leading to more efficient and accurate solutions. The focus is increasingly on handling ill-posed problems, which are common in medical imaging, tomography, and other high-dimensional inverse estimation tasks.
One of the key trends is the use of generative models, particularly diffusion models, to address the challenges posed by complex priors and high-dimensional parameter spaces. These models are being combined with physics-informed neural networks (PINNs) to enhance the generalization and computational efficiency of the solutions. The incorporation of stochastic differential equations (SDEs) and particle filtering techniques is also gaining traction, enabling a more thorough exploration of the solution space and improving the robustness of the inverse solvers.
Another notable development is the application of subspace-based dimension reduction techniques to accelerate the sampling process in nonlinear PDE-based inverse problems. This approach allows for faster and more accurate reconstructions by reducing the computational burden associated with high-dimensional data. Additionally, there is a growing interest in developing methods that can handle batch updates and parallel computational schemes, which are crucial for real-time applications in fields like biomedical imaging and tomography.
Noteworthy Innovations
Particle-Filtering-based Latent Diffusion for Inverse Problems: This approach introduces a novel framework for nonlinear exploration of the solution space, significantly outperforming state-of-the-art methods in tasks like super-resolution, deblurring, and inpainting.
Batch-FPM: Random batch-update multi-parameter physical Fourier ptychography neural network: This method achieves near real-time digital refocusing with improved noise resistance and reconstruction speed, making it highly applicable in clinical diagnostics and biomedical research.
A Score-based Generative Solver for PDE-constrained Inverse Problems with Complex Priors: The proposed score-based diffusion model, combined with a physics-informed CNN surrogate, effectively learns geometrical features from prior samples, leading to superior inverse estimation results.
Subspace Diffusion Posterior Sampling for Travel-Time Tomography: This work demonstrates significant advancements in improving travel-time imaging quality and reducing sampling time for reconstruction, leveraging subspace-based dimension reduction techniques.
