The Fine-Grained Complexity of Problems Expressible by First-Order Logic and Its Extensions

This dissertation studies the fine-grained complexity of model checking problems for fixed logical formulas on sparse input structures. The Orthogonal Vectors problem is an important and well-studied problem in fine-grained complexity: its hardness is implied by the Strong Exponential Time Hypothesis, and its hardness implies the hardness of many other interesting problems. We show that the Orthogonal Vectors problem is complete in the class of first-order model checking on sparse structures, under fine-grained reductions. In other words, the hardness of Orthogonal Vectors and the hardness of first-order model checking imply each other. This also gives us an improved algorithm for first-order model checking problems. Among all first-order logic formulas in prenex normal form, we have reasons to believe that quantifier structures $\exists \dots \exists \forall$ and $\forall \dots \forall \exists$ may be the hardest in computational complexity: If the Nondeterministic version of the Strong Exponential Time Hypothesis is true, formulas of these forms are the only hard ones under the Strong Exponential Time Hypothesis. We can add extensions to first-order logic to strengthen its expressive power. This work also studies the fine-grained complexity of first-order formulas with comparison on structures with total order, first-order formulas with transitive closure operations, first-order formulas of fixed quantifier rank, and first-order formulas of fixed variable complexity. We also introduce a technique that can be used to reduce from sequential problems on graphs to parallel problems on sets, which can be applied to extending the Least Weight Subsequence problems from linear structures to some special classes of graphs

A deep learning framework for multi-scale models based on physics-informed neural networks

Physics-informed neural networks (PINN) combine deep neural networks with the solution of partial differential equations (PDEs), creating a new and promising research area for numerically solving PDEs. Faced with a class of multi-scale problems that include loss terms of different orders of magnitude in the loss function, it is challenging for standard PINN methods to obtain an available prediction. In this paper, we propose a new framework for solving multi-scale problems by reconstructing the loss function. The framework is based on the standard PINN method, and it modifies the loss function of the standard PINN method by applying different numbers of power operations to the loss terms of different magnitudes, so that the individual loss terms composing the loss function have approximately the same order of magnitude among themselves. In addition, we give a grouping regularization strategy, and this strategy can deal well with the problem which varies significantly in different subdomains. The proposed method enables loss terms with different magnitudes to be optimized simultaneously, and it advances the application of PINN for multi-scale problems

PSP: Pre-trained Soft Prompts for Few-Shot Abstractive Summarization

Few-shot abstractive summarization has become a challenging task in natural language generation. To support it, we designed a novel soft prompts architecture coupled with a prompt pre-training plus fine-tuning paradigm that is effective and tunes only extremely light parameters. The soft prompts include continuous input embeddings across an encoder and a decoder to fit the structure of the generation models. Importantly, a novel inner-prompt placed in the text is introduced to capture document-level information. The aim is to devote attention to understanding the document that better prompts the model to generate document-related content. The first step in the summarization procedure is to conduct prompt pre-training with self-supervised pseudo-data. This teaches the model basic summarizing capabilities. The model is then fine-tuned with few-shot examples. Experimental results on the CNN/DailyMail and XSum datasets show that our method, with only 0.1% of the parameters, outperforms full-model tuning where all model parameters are tuned. It also surpasses Prompt Tuning by a large margin and delivers competitive results against Prefix-Tuning with 3% of the parameters.Comment: 12 page