Sidecar: A Structure-Preserving Framework for Solving Partial Differential Equations with Neural Networks

Solving partial differential equations (PDEs) with neural networks (NNs) has shown great potential in various scientific and engineering fields. However, most existing NN solvers mainly focus on satisfying the given PDEs, without explicitly considering intrinsic physical properties such as mass conservation or energy dissipation. This limitation can result in unstable or nonphysical solutions, particularly in long-term simulations. To address this issue, we propose Sidecar, a novel framework that enhances the accuracy and physical consistency of existing NN solvers by incorporating structure-preserving knowledge. Inspired by the Time-Dependent Spectral Renormalization (TDSR) approach, our Sidecar framework introduces a small copilot network, which is trained to guide the existing NN solver in preserving physical structure. This framework is designed to be highly flexible, enabling the incorporation of structure-preserving principles from diverse PDEs into a wide range of NN solvers. Our experimental results on benchmark PDEs demonstrate the improvement of the existing neural network solvers in terms of accuracy and consistency with structure-preserving properties.

Neural Networks Trained by Weight Permutation are Universal Approximators

The universal approximation property is fundamental to the success of neural networks, and has traditionally been achieved by training networks without any constraints on their parameters. However, recent experimental research proposed a novel permutation-based training method, which exhibited a desired classification performance without modifying the exact weight values. In this paper, we provide a theoretical guarantee of this permutation training method by proving its ability to guide a ReLU network to approximate one-dimensional continuous functions. Our numerical results further validate this method's efficiency in regression tasks with various initializations. The notable observations during weight permutation suggest that permutation training can provide an innovative tool for describing network learning behavior.

Inverse Evolution Layers: Physics-informed Regularizers for Deep Neural Networks

This paper proposes a novel approach to integrating partial differential equation (PDE)-based evolution models into neural networks through a new type of regularization. Specifically, we propose inverse evolution layers (IELs) based on evolution equations. These layers can achieve specific regularization objectives and endow neural networks' outputs with corresponding properties of the evolution models. Moreover, IELs are straightforward to construct and implement, and can be easily designed for various physical evolutions and neural networks. Additionally, the design process for these layers can provide neural networks with intuitive and mathematical interpretability, thus enhancing the transparency and explainability of the approach. To demonstrate the effectiveness, efficiency, and simplicity of our approach, we present an example of endowing semantic segmentation models with the smoothness property based on the heat diffusion model. To achieve this goal, we design heat-diffusion IELs and apply them to address the challenge of semantic segmentation with noisy labels. The experimental results demonstrate that the heat-diffusion IELs can effectively mitigate the overfitting problem caused by noisy labels.

An Active Contour Model with Local Variance Force Term and Its Efficient Minimization Solver for Multi-phase Image Segmentation

In this paper, we propose an active contour model with a local variance force (LVF) term that can be applied to multi-phase image segmentation problems. With the LVF, the proposed model is very effective in the segmentation of images with noise. To solve this model efficiently, we represent the regularization term by characteristic functions and then design a minimization algorithm based on a modification of the iterative convolution-thresholding method (ICTM), namely ICTM-LVF. This minimization algorithm enjoys the energy-decaying property under some conditions and has highly efficient performance in the segmentation. To overcome the initialization issue of active contour models, we generalize the inhomogeneous graph Laplacian initialization method (IGLIM) to the multi-phase case and then apply it to give the initial contour of the ICTM-LVF solver. Numerical experiments are conducted on synthetic images and real images to demonstrate the capability of our initialization method, and the effectiveness of the local variance force for noise robustness in the multi-phase image segmentation.