Unisolver: PDE-Conditional Transformers Are Universal PDE Solvers

Deep models have recently emerged as a promising tool to solve partial differential equations (PDEs), known as neural PDE solvers. While neural solvers trained from either simulation data or physics-informed loss can solve the PDEs reasonably well, they are mainly restricted to a specific set of PDEs, e.g. a certain equation or a finite set of coefficients. This bottleneck limits the generalizability of neural solvers, which is widely recognized as its major advantage over numerical solvers. In this paper, we present the Universal PDE solver (Unisolver) capable of solving a wide scope of PDEs by leveraging a Transformer pre-trained on diverse data and conditioned on diverse PDEs. Instead of simply scaling up data and parameters, Unisolver stems from the theoretical analysis of the PDE-solving process. Our key finding is that a PDE solution is fundamentally under the control of a series of PDE components, e.g. equation symbols, coefficients, and initial and boundary conditions. Inspired by the mathematical structure of PDEs, we define a complete set of PDE components and correspondingly embed them as domain-wise (e.g. equation symbols) and point-wise (e.g. boundaries) conditions for Transformer PDE solvers. Integrating physical insights with recent Transformer advances, Unisolver achieves consistent state-of-the-art results on three challenging large-scale benchmarks, showing impressive gains and endowing favorable generalizability and scalability.

Properties on n-dimensional convolution for image deconvolution

Convolution system is linear and time invariant, and can describe the optical imaging process. Based on convolution system, many deconvolution techniques have been developed for optical image analysis, such as boosting the space resolution of optical images, image denoising, image enhancement and so on. Here, we gave properties on N-dimensional convolution. By using these properties, we proposed image deconvolution method. This method uses a series of convolution operations to deconvolute image. We demonstrated that the method has the similar deconvolution results to the state-of-art method. The core calculation of the proposed method is image convolution, and thus our method can easily be integrated into GPU mode for large-scale image deconvolution.