A DeepONet joint Neural Tangent Kernel Hybrid Framework for Physics-Informed Inverse Source Problems and Robust Image Reconstruction

TL;DR

This paper addresses inverse source localization and image reconstruction under governing Navier–Stokes physics using a novel DeepONet–NTK hybrid framework. The approach blends operator learning with Neural Tangent Kernel–based training stabilization and a physics-informed, task-specific loss, enabling robust performance with limited or noisy data. Key contributions include integrating DeepONet with NTK for improved convergence and generalization, incorporating physics-informed and perceptual losses, and validating on synthetic NS data and standard image datasets with strong quantitative and qualitative results. The work demonstrates potential for scalable, physically consistent solutions in computational physics and imaging applications, with implications for real-world inverse problems and robust image restoration.

Abstract

This work presents a novel hybrid approach that integrates Deep Operator Networks (DeepONet) with the Neural Tangent Kernel (NTK) to solve complex inverse problem. The method effectively addresses tasks such as source localization governed by the Navier-Stokes equations and image reconstruction, overcoming challenges related to nonlinearity, sparsity, and noisy data. By incorporating physics-informed constraints and task-specific regularization into the loss function, the framework ensures solutions that are both physically consistent and accurate. Validation on diverse synthetic and real datasets demonstrates its robustness, scalability, and precision, showcasing its broad potential applications in computational physics and imaging sciences.

Figures (4)

• Figure 1: The detailed structure of the DeepONet-NTKS model is presented, illustrating the training process of the DeepONet + NTK framework. On the left, the Branch and Trunk Networks are used to extract features from the input function $u$ and location $y$, generating the feature vectors $\mathbf{b}$ and $\mathbf{t}$. In the middle, the NTK component incorporates kernel similarities $\Theta_{ij}(x, x')$ to facilitate feature fusion. On the right, the loss function combines Mean Squared Error (MSE) with NTK-based regularization $L_{\text{NTK}}$, optimizing the model parameters $\theta^*$ to enhance prediction accuracy and stability. This figure illustrates the integration of NTK regularization into DeepONet for improved performance and robustness.
• Figure 2: Quantitative comparison of five evaluation metrics (PSNR, SSIM, and MSE) across various methods on CIFAR-10, CIFAR-100, MNIST, and Fashion-MNIST datasets. Higher PSNR and SSIM values, as well as lower MSE values, indicate better performance. This figure highlights the comparative effectiveness of the evaluated methods.
• Figure 3: (a) Source distribution within the solution domain of the Navier-Stokes equation. This subfigure illustrates randomly placed point sources used to generate synthetic data, with varying source locations to simulate different fluid dynamics scenarios. (b) Comparison of true and predicted source locations for the Navier-Stokes inverse source problem. This subfigure demonstrates the high accuracy of the model in predicting source positions by comparing predicted locations with the true ones.
• Figure 4: Reconstruction results on the (a) MNIST, (b) CIFAR-10, (c) CIFAR-100, and (d) FashionMNIST datasets. The top row displays the ground truth images, the middle row shows the corrupted images, and the bottom row presents the reconstructed images generated by the proposed method. This figure illustrates the effectiveness of the proposed method in reconstructing images across different datasets. It highlights how the method restores corrupted inputs to closely resemble the ground truth, demonstrating its robustness and generalizability across diverse datasets with varying levels of complexity.