Neural Networks as Regularizers: A New Framework for Solving Ill-Posed Inverse Problems

In the realm of scientific computing, inverse problems—where one reconstructs hidden parameters from indirect, noisy measurements—are notoriously challenging. These issues pervade fields like medical imaging and geophysics, where traditional s often falter due to computational inefficiency and noise sensitivity. A groundbreaking study by researchers from Beijing Institute of Technology and The Chinese University of Hong Kong introduces neural networks as a robust regularization tool, proving they can stabilize solutions for general nonlinear ill-posed operator equations. By establishing universal approximation theorems that account for both approximation and measurement errors, the work shifts the paradigm from handcrafted priors to data-driven neural architectures, offering a fresh perspective on solving equations like A(f) = g with noisy data gδ.

Ologically, the paper proposes two innovative approaches: the expanding neural network and a neural network-based Tikhonov regularization scheme. The expanding iteratively increases the number of neurons in a two-layer network, using the neuron count as both a regularization parameter and iteration number, guided by the Morozov discrepancy principle to halt when the residual error falls below a threshold. For Tikhonov regularization, the framework minimizes a functional combining data fidelity and a penalty term based on the Barron norm, ensuring stability through an explicit regularization parameter α. Both s leverage the Barron space properties, where functions admit representations via probability measures, enabling efficient approximation with ReLU-activated networks and rigorous error analysis under conditions like local Hölder continuity of the operator A.

From the study demonstrate that neural networks can achieve stable, convergent solutions even in high-noise scenarios. For instance, in the expanding , the number of neurons scales as O(δ^(-2/θ)) with noise level δ and Hölder exponent θ, meaning that smaller networks suffice for noisier data to avoid overfitting. Numerical experiments on 1D Fredholm integral equations, auto-convolution Volterra equations, and 2D electrical impedance tomography show that relative L2 errors decrease with neuron count up to a point, after which instability can arise without proper regularization. The Tikhonov approach, in particular, suppresses blow-up behaviors observed in iterative s, highlighting its robustness in balancing accuracy and stability across various random seeds and noise levels.

Of this research are profound for applied fields reliant on inverse problems, such as medical imaging and computational physics. By providing theoretical guarantees for neural network regularizers, it enables more efficient and reliable reconstructions in tasks like MRI or CT scanning, where noise and computational limits are critical. the notion that bigger networks are always better, advocating for adaptive sizing that aligns with noise characteristics, which could lead to optimized AI-driven tools in healthcare and engineering. Moreover, the convergence rates derived under variational source conditions offer a roadmap for achieving optimal performance, potentially accelerating innovations in real-time data processing and inverse problem solvers.

Despite its strengths, the study has limitations, primarily its focus on shallow two-layer networks, which restricts the exploitation of depth-related efficiencies seen in deep learning. Extending the analysis to multi-layer architectures remains an open due to the lack of uniform parameter bounds for deeper networks. Additionally, the non-convex optimization problems involved pose practical hurdles, as efficient algorithms for minimizing the proposed functionals are not fully addressed. Future work could explore graph convolutional neural networks or deeper Barron-type spaces to enhance applicability, but for now, this research solidifies neural networks as a viable regularization framework with rigorous mathematical backing.

Reference: Wang et al., 2025, arXiv:2511.16171v1