Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics, and explores how to make it more robust--and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than classical feed-forward neural networks, recurrent neural networks, and convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering, In this work, we review such methods and extend them for parametric studies as well as for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications.

Blind image quality assessment (BIQA) remains challenging due to the diversity of distortion and image content variation, which complicate the distortion patterns crossing different scales and aggravate the difficulty of the regression problem for BIQA. However, existing BIQA methods often fail to consider multi-scale distortion patterns and image content, and little research has been done on learning strategies to make the regression model produce better performance. In this paper, we propose a simple yet effective Progressive Multi-Task Image Quality Assessment (PMT-IQA) model, which contains a multi-scale feature extraction module (MS) and a progressive multi-task learning module (PMT), to help the model learn complex distortion patterns and better optimize the regression issue to align with the law of human learning process from easy to hard. To verify the effectiveness of the proposed PMT-IQA model, we conduct experiments on four widely used public datasets, and the experimental results indicate that the performance of PMT-IQA is superior to the comparison approaches, and both MS and PMT modules improve the model's performance.

With the development of technology and sharing economy, Airbnb as a famous short-term rental platform, has become the first choice for many young people to select. The issue of Airbnb's pricing has always been a problem worth studying. While the previous studies achieve promising results, there are exists deficiencies to solve. Such as, (1) the feature attributes of rental are not rich enough; (2) the research on rental text information is not deep enough; (3) there are few studies on predicting the rental price combined with the point of interest(POI) around the house. To address the above challenges, we proposes a multi-source information embedding(MSIE) model to predict the rental price of Airbnb. Specifically, we first selects the statistical feature to embed the original rental data. Secondly, we generates the word feature vector and emotional score combination of three different text information to form the text feature embedding. Thirdly, we uses the points of interest(POI) around the rental house information generates a variety of spatial network graphs, and learns the embedding of the network to obtain the spatial feature embedding. Finally, this paper combines the three modules into multi source rental representations, and uses the constructed fully connected neural network to predict the price. The analysis of the experimental results shows the effectiveness of our proposed model.

Neural operators, which emerge as implicit solution operators of hidden governing equations, have recently become popular tools for learning responses of complex real-world physical systems. Nevertheless, the majority of neural operator applications has thus far been data-driven, which neglects the intrinsic preservation of fundamental physical laws in data. In this paper, we introduce a novel integral neural operator architecture, to learn physical models with fundamental conservation laws automatically guaranteed. In particular, by replacing the frame-dependent position information with its invariant counterpart in the kernel space, the proposed neural operator is by design translation- and rotation-invariant, and consequently abides by the conservation laws of linear and angular momentums. As applications, we demonstrate the expressivity and efficacy of our model in learning complex material behaviors from both synthetic and experimental datasets, and show that, by automatically satisfying these essential physical laws, our learned neural operator is not only generalizable in handling translated and rotated datasets, but also achieves state-of-the-art accuracy and efficiency as compared to baseline neural operator models.

With the development of natural language processing techniques(NLP), automatic diagnosis of eye diseases using ophthalmology electronic medical records (OEMR) has become possible. It aims to evaluate the condition of both eyes of a patient respectively, and we formulate it as a particular multi-label classification task in this paper. Although there are a few related studies in other diseases, automatic diagnosis of eye diseases exhibits unique characteristics. First, descriptions of both eyes are mixed up in OEMR documents, with both free text and templated asymptomatic descriptions, resulting in sparsity and clutter of information. Second, OEMR documents contain multiple parts of descriptions and have long document lengths. Third, it is critical to provide explainability to the disease diagnosis model. To overcome those challenges, we present an effective automatic eye disease diagnosis framework, NEEDED. In this framework, a preprocessing module is integrated to improve the density and quality of information. Then, we design a hierarchical transformer structure for learning the contextualized representations of each sentence in the OEMR document. For the diagnosis part, we propose an attention-based predictor that enables traceable diagnosis by obtaining disease-specific information. Experiments on the real dataset and comparison with several baseline models show the advantage and explainability of our framework.

Practical applications employing deep learning must guarantee inference quality. However, we found that the inference quality of state-of-the-art and state-of-the-practice in practical applications has a long tail distribution. In the real world, many tasks have strict requirements for the quality of deep learning inference, such as safety-critical and mission-critical tasks. The fluctuation of inference quality seriously affects its practical applications, and the quality at the tail may lead to severe consequences. State-of-the-art and state-of-the-practice with outstanding inference quality designed and trained under loose constraints still have poor inference quality under constraints with practical application significance. On the one hand, the neural network models must be deployed on complex systems with limited resources. On the other hand, safety-critical and mission-critical tasks need to meet more metric constraints while ensuring high inference quality. We coin a new term, ``tail quality,'' to characterize this essential requirement and challenge. We also propose a new metric, ``X-Critical-Quality,'' to measure the inference quality under certain constraints. This article reveals factors contributing to the failure of using state-of-the-art and state-of-the-practice algorithms and systems in real scenarios. Therefore, we call for establishing innovative methodologies and tools to tackle this enormous challenge.

We present X-Decoder, a generalized decoding model that can predict pixel-level segmentation and language tokens seamlessly. X-Decodert takes as input two types of queries: (i) generic non-semantic queries and (ii) semantic queries induced from text inputs, to decode different pixel-level and token-level outputs in the same semantic space. With such a novel design, X-Decoder is the first work that provides a unified way to support all types of image segmentation and a variety of vision-language (VL) tasks. Further, our design enables seamless interactions across tasks at different granularities and brings mutual benefits by learning a common and rich pixel-level visual-semantic understanding space, without any pseudo-labeling. After pretraining on a mixed set of a limited amount of segmentation data and millions of image-text pairs, X-Decoder exhibits strong transferability to a wide range of downstream tasks in both zero-shot and finetuning settings. Notably, it achieves (1) state-of-the-art results on open-vocabulary segmentation and referring segmentation on eight datasets; (2) better or competitive finetuned performance to other generalist and specialist models on segmentation and VL tasks; and (3) flexibility for efficient finetuning and novel task composition (e.g., referring captioning and image editing). Code, demo, video, and visualization are available at https://x-decoder-vl.github.io.

Inductive reasoning is a core component of human intelligence. In the past research of inductive reasoning within computer science, logic language is used as representations of knowledge (facts and rules, more specifically). However, logic language can cause systematic problems for inductive reasoning such as disability of handling raw input such as natural language, sensitiveness to mislabeled data, and incapacity to handle ambiguous input. To this end, we propose a new task, which is to induce natural language rules from natural language facts, and create a dataset termed DEER containing 1.2k rule-fact pairs for the task, where rules and facts are written in natural language. New automatic metrics are also proposed and analysed for the evaluation of this task. With DEER, we investigate a modern approach for inductive reasoning where we use natural language as representation for knowledge instead of logic language and use pretrained language models as ''reasoners''. Moreover, we provide the first and comprehensive analysis of how well pretrained language models can induce natural language rules from natural language facts. We also propose a new framework drawing insights from philosophy literature for this task, which we show in the experiment section that surpasses baselines in both automatic and human evaluations.

Previous computation models either have equivalent abilities in representing all computations but fail to provide primitive operators for programming complex algorithms or lack generalized expression ability to represent newly-added computations. This article presents a unified computation model with generalized expression ability and a concise set of primitive operators for programming high-level algorithms. We propose a unified data abstraction -- Tensor of List, and offer a unified computation model based on Tensor of List, which we call the ToL model (in short, ToL). ToL introduces five atomic computations that can represent any elementary computation by finite composition, ensured with strict formal proof. Based on ToL, we design a pure-functional language -- ToLang. ToLang provides a concise set of primitive operators that can be used to program complex big data and AI algorithms. Our evaluations show ToL has generalized expression ability and a built-in performance indicator, born with a strictly defined computation metric -- elementary operation count (EOPs), consistent with FLOPs within a small error range.

Structured tabular data exist across nearly all fields. Reasoning task over these data aims to answer questions or determine the truthiness of hypothesis sentences by understanding the semantic meaning of a table. While previous works have devoted significant efforts to the tabular reasoning task, they always assume there are sufficient labeled data. However, constructing reasoning samples over tables (and related text) is labor-intensive, especially when the reasoning process is complex. When labeled data is insufficient, the performance of models will suffer an unendurable decline. In this paper, we propose a unified framework for unsupervised complex tabular reasoning (UCTR), which generates sufficient and diverse synthetic data with complex logic for tabular reasoning tasks, assuming no human-annotated data at all. We first utilize a random sampling strategy to collect diverse programs of different types and execute them on tables based on a "Program-Executor" module. To bridge the gap between the programs and natural language sentences, we design a powerful "NL-Generator" module to generate natural language sentences with complex logic from these programs. Since a table often occurs with its surrounding texts, we further propose novel "Table-to-Text" and "Text-to-Table" operators to handle joint table-text reasoning scenarios. This way, we can adequately exploit the unlabeled table resources to obtain a well-performed reasoning model under an unsupervised setting. Our experiments cover different tasks (question answering and fact verification) and different domains (general and specific), showing that our unsupervised methods can achieve at most 93% performance compared to supervised models. We also find that it can substantially boost the supervised performance in low-resourced domains as a data augmentation technique. Our code is available at https://github.com/leezythu/UCTR.