In the field of medical image compression, Implicit Neural Representation (INR) networks have shown remarkable versatility due to their flexible compression ratios, yet they are constrained by a one-to-one fitting approach that results in lengthy encoding times. Our novel method, ``\textbf{UniCompress}'', innovatively extends the compression capabilities of INR by being the first to compress multiple medical data blocks using a single INR network. By employing wavelet transforms and quantization, we introduce a codebook containing frequency domain information as a prior input to the INR network. This enhances the representational power of INR and provides distinctive conditioning for different image blocks. Furthermore, our research introduces a new technique for the knowledge distillation of implicit representations, simplifying complex model knowledge into more manageable formats to improve compression ratios. Extensive testing on CT and electron microscopy (EM) datasets has demonstrated that UniCompress outperforms traditional INR methods and commercial compression solutions like HEVC, especially in complex and high compression scenarios. Notably, compared to existing INR techniques, UniCompress achieves a 4$\sim$5 times increase in compression speed, marking a significant advancement in the field of medical image compression. Codes will be publicly available.

Solving partial differential equations (PDEs) has been a fundamental problem in computational science and of wide applications for both scientific and engineering research. Due to its universal approximation property, neural network is widely used to approximate the solutions of PDEs. However, existing works are incapable of solving high-order PDEs due to insufficient calculation accuracy of higher-order derivatives, and the final network is a black box without explicit explanation. To address these issues, we propose a deep learning framework to solve high-order PDEs, named SHoP. Specifically, we derive the high-order derivative rule for neural network, to get the derivatives quickly and accurately; moreover, we expand the network into a Taylor series, providing an explicit solution for the PDEs. We conduct experimental validations four high-order PDEs with different dimensions, showing that we can solve high-order PDEs efficiently and accurately.

Causal discovery from time-series data has been a central task in machine learning. Recently, Granger causality inference is gaining momentum due to its good explainability and high compatibility with emerging deep neural networks. However, most existing methods assume structured input data and degenerate greatly when encountering data with randomly missing entries or non-uniform sampling frequencies, which hampers their applications in real scenarios. To address this issue, here we present CUTS, a neural Granger causal discovery algorithm to jointly impute unobserved data points and build causal graphs, via plugging in two mutually boosting modules in an iterative framework: (i) Latent data prediction stage: designs a Delayed Supervision Graph Neural Network (DSGNN) to hallucinate and register unstructured data which might be of high dimension and with complex distribution; (ii) Causal graph fitting stage: builds a causal adjacency matrix with imputed data under sparse penalty. Experiments show that CUTS effectively infers causal graphs from unstructured time-series data, with significantly superior performance to existing methods. Our approach constitutes a promising step towards applying causal discovery to real applications with non-ideal observations.