Zecheng Zhang

Hao Liu, Zecheng Zhang, Wenjing Liao et al.

Neural scaling laws play a pivotal role in the performance of deep neural networks and have been observed in a wide range of tasks. However, a complete theoretical framework for understanding these scaling laws remains underdeveloped. In this paper, we explore the neural scaling laws for deep operator networks, which involve learning mappings between function spaces, with a focus on the Chen and Chen style architecture. These approaches, which include the popular Deep Operator Network (DeepONet), approximate the output functions using a linear combination of learnable basis functions and coefficients that depend on the input functions. We establish a theoretical framework to quantify the neural scaling laws by analyzing its approximation and generalization errors. We articulate the relationship between the approximation and generalization errors of deep operator networks and key factors such as network model size and training data size. Moreover, we address cases where input functions exhibit low-dimensional structures, allowing us to derive tighter error bounds. These results also hold for deep ReLU networks and other similar structures. Our results offer a partial explanation of the neural scaling laws in operator learning and provide a theoretical foundation for their applications.

Tianqi Xu, Linyao Chen, Dai-Jie Wu et al.

The development of autonomous agents increasingly relies on Multimodal Language Models (MLMs) to perform tasks described in natural language with GUI environments, such as websites, desktop computers, or mobile phones. Existing benchmarks for MLM agents in interactive environments are limited by their focus on a single environment, lack of detailed and generalized evaluation methods, and the complexities of constructing tasks and evaluators. To overcome these limitations, we introduce Crab, the first agent benchmark framework designed to support cross-environment tasks, incorporating a graph-based fine-grained evaluation method and an efficient mechanism for task and evaluator construction. Our framework supports multiple devices and can be easily extended to any environment with a Python interface. Leveraging Crab, we developed a cross-platform Crab Benchmark-v0 comprising 120 tasks in computer desktop and mobile phone environments. We evaluated four advanced MLMs using different single and multi-agent system configurations on this benchmark. The experimental results demonstrate that the single agent with GPT-4o achieves the best completion ratio of 38.01%. All framework code, agent code, and task datasets are publicly available at https://github.com/camel-ai/crab.

Bowen Song, Soo Min Kwon, Zecheng Zhang et al.

Diffusion models have recently emerged as powerful generative priors for solving inverse problems. However, training diffusion models in the pixel space are both data-intensive and computationally demanding, which restricts their applicability as priors for high-dimensional real-world data such as medical images. Latent diffusion models, which operate in a much lower-dimensional space, offer a solution to these challenges. However, incorporating latent diffusion models to solve inverse problems remains a challenging problem due to the nonlinearity of the encoder and decoder. To address these issues, we propose \textit{ReSample}, an algorithm that can solve general inverse problems with pre-trained latent diffusion models. Our algorithm incorporates data consistency by solving an optimization problem during the reverse sampling process, a concept that we term as hard data consistency. Upon solving this optimization problem, we propose a novel resampling scheme to map the measurement-consistent sample back onto the noisy data manifold and theoretically demonstrate its benefits. Lastly, we apply our algorithm to solve a wide range of linear and nonlinear inverse problems in both natural and medical images, demonstrating that our approach outperforms existing state-of-the-art approaches, including those based on pixel-space diffusion models.

Joshua Robinson, Rishabh Ranjan, Weihua Hu et al.

We present RelBench, a public benchmark for solving predictive tasks over relational databases with graph neural networks. RelBench provides databases and tasks spanning diverse domains and scales, and is intended to be a foundational infrastructure for future research. We use RelBench to conduct the first comprehensive study of Relational Deep Learning (RDL) (Fey et al., 2024), which combines graph neural network predictive models with (deep) tabular models that extract initial entity-level representations from raw tables. End-to-end learned RDL models fully exploit the predictive signal encoded in primary-foreign key links, marking a significant shift away from the dominant paradigm of manual feature engineering combined with tabular models. To thoroughly evaluate RDL against this prior gold-standard, we conduct an in-depth user study where an experienced data scientist manually engineers features for each task. In this study, RDL learns better models whilst reducing human work needed by more than an order of magnitude. This demonstrates the power of deep learning for solving predictive tasks over relational databases, opening up many new research opportunities enabled by RelBench.

Zecheng Zhang, Christian Moya, Lu Lu et al.

Neural operators have been applied in various scientific fields, such as solving parametric partial differential equations, dynamical systems with control, and inverse problems. However, challenges arise when dealing with input functions that exhibit heterogeneous properties, requiring multiple sensors to handle functions with minimal regularity. To address this issue, discretization-invariant neural operators have been used, allowing the sampling of diverse input functions with different sensor locations. However, existing frameworks still require an equal number of sensors for all functions. In our study, we propose a novel distributed approach to further relax the discretization requirements and solve the heterogeneous dataset challenges. Our method involves partitioning the input function space and processing individual input functions using independent and separate neural networks. A centralized neural network is used to handle shared information across all output functions. This distributed methodology reduces the number of gradient descent back-propagation steps, improving efficiency while maintaining accuracy. We demonstrate that the corresponding neural network is a universal approximator of continuous nonlinear operators and present four numerical examples to validate its performance.

Yuxuan Liu, Zecheng Zhang, Hayden Schaeffer

Approximating nonlinear differential equations using a neural network provides a robust and efficient tool for various scientific computing tasks, including real-time predictions, inverse problems, optimal controls, and surrogate modeling. Previous works have focused on embedding dynamical systems into networks through two approaches: learning a single solution operator (i.e., the mapping from input parametrized functions to solutions) or learning the governing system of equations (i.e., the constitutive model relative to the state variables). Both of these approaches yield different representations for the same underlying data or function. Additionally, observing that families of differential equations often share key characteristics, we seek one network representation across a wide range of equations. Our method, called Predicting Operators and Symbolic Expressions (PROSE), learns maps from multimodal inputs to multimodal outputs, capable of generating both numerical predictions and mathematical equations. By using a transformer structure and a feature fusion approach, our network can simultaneously embed sets of solution operators for various parametric differential equations using a single trained network. Detailed experiments demonstrate that the network benefits from its multimodal nature, resulting in improved prediction accuracy and better generalization. The network is shown to be able to handle noise in the data and errors in the symbolic representation, including noisy numerical values, model misspecification, and erroneous addition or deletion of terms. PROSE provides a new neural network framework for differential equations which allows for more flexibility and generality in learning operators and governing equations from data.

Yuxuan Liu, Jingmin Sun, Xinjie He et al.

We propose PROSE-FD, a zero-shot multimodal PDE foundational model for simultaneous prediction of heterogeneous two-dimensional physical systems related to distinct fluid dynamics settings. These systems include shallow water equations and the Navier-Stokes equations with incompressible and compressible flow, regular and complex geometries, and different buoyancy settings. This work presents a new transformer-based multi-operator learning approach that fuses symbolic information to perform operator-based data prediction, i.e. non-autoregressive. By incorporating multiple modalities in the inputs, the PDE foundation model builds in a pathway for including mathematical descriptions of the physical behavior. We pre-train our foundation model on 6 parametric families of equations collected from 13 datasets, including over 60K trajectories. Our model outperforms popular operator learning, computer vision, and multi-physics models, in benchmark forward prediction tasks. We test our architecture choices with ablation studies.

Jingmin Sun, Zecheng Zhang, Hayden Schaeffer

Single-operator learning involves training a deep neural network to learn a specific operator, whereas recent work in multi-operator learning uses an operator embedding structure to train a single neural network on data from multiple operators. Thus, multi-operator learning is capable of predicting a range of operators within one model. In this work, we propose pretraining and fine-tuning strategies for solving PDEs using multi-operator learning. One key aspect is that by increasing the number of families of operators used in pretraining, a PDE foundation model can be fine-tuned to downstream tasks involving new PDEs with a limited number of samples, thus outperforming single operator neural networks. Specifically, a multi-operator learning model pre-trained with data from diverse PDE families can predict unseen operators after fine-tuning with only a limited number of operators from the new family, enabling them to serve as a data-free PDE solver. We also show that the proposed training and fine-tuning method is able to predict new operators in zero-shot prediction without samples. Additionally, we introduce a PDE-agnostic meta-learning algorithm to improve the adaptability of the model to various PDEs by providing a better parameter initialization process. To address the needs of applications with limited computing resources, we explore low-rank adaptation methods that reduce computational costs while enhancing solver accuracy. Lastly, by examining the scaling law with respect to the number of operator families, we establish and highlight its potential for broad adaptation in PDE-solving tasks.

Derek Jollie, Jingmin Sun, Zecheng Zhang et al.

Symbolic encoding has been used in multi-operator learning as a way to embed additional information for distinct time-series data. For spatiotemporal systems described by time-dependent partial differential equations, the equation itself provides an additional modality to identify the system. The utilization of symbolic expressions along side time-series samples allows for the development of multimodal predictive neural networks. A key challenge with current approaches is that the symbolic information, i.e. the equations, must be manually preprocessed (simplified, rearranged, etc.) to match and relate to the existing token library, which increases costs and reduces flexibility, especially when dealing with new differential equations. We propose a new token library based on SymPy to encode differential equations as an additional modality for time-series models. The proposed approach incurs minimal cost, is automated, and maintains high prediction accuracy for forecasting tasks. Additionally, we include a Bayesian filtering module that connects the different modalities to refine the learned equation. This improves the accuracy of the learned symbolic representation and the predicted time-series.

Zecheng Zhang

Grid startup, an integral component of the power system, holds strategic importance for ensuring the reliability and efficiency of the electrical grid. However, current methodologies for in-depth analysis and precise prediction of grid startup scenarios are inadequate. To address these challenges, we propose a novel method based on the Transformer-LSTM-PSO model. This model uniquely combines the Transformer's self-attention mechanism, LSTM's temporal modeling capabilities, and the parameter tuning features of the particle swarm optimization algorithm. It is designed to more effectively capture the complex temporal relationships in grid startup schemes. Our experiments demonstrate significant improvements, with our model achieving lower RMSE and MAE values across multiple datasets compared to existing benchmarks, particularly in the NYISO Electric Market dataset where the RMSE was reduced by approximately 15% and the MAE by 20% compared to conventional models. Our main contribution is the development of a Transformer-LSTM-PSO model that significantly enhances the accuracy and efficiency of smart grid startup predictions. The application of the Transformer-LSTM-PSO model represents a significant advancement in smart grid predictive analytics, concurrently fostering the development of more reliable and intelligent grid management systems.

Min Zhu, Jingmin Sun, Zecheng Zhang et al.

Partial differential equations (PDEs) govern a wide range of physical systems, and recent multimodal foundation models have shown promise for learning PDE solution operators across diverse equation families. However, existing multi-operator learning approaches are data-hungry and neglect physics during training. Here, we propose a physics-informed multimodal foundation model (PI-MFM) framework that directly enforces governing equations during pretraining and adaptation. PI-MFM takes symbolic representations of PDEs as the input, and automatically assembles PDE residual losses from the input expression via a vectorized derivative computation. These designs enable any PDE-encoding multimodal foundation model to be trained or adapted with unified physics-informed objectives across equation families. On a benchmark of 13 parametric one-dimensional time-dependent PDE families, PI-MFM consistently outperforms purely data-driven counterparts, especially with sparse labeled spatiotemporal points, partially observed time domains, or few labeled function pairs. Physics losses further improve robustness against noise, and simple strategies such as resampling collocation points substantially improve accuracy. We also analyze the accuracy, precision, and computational cost of automatic differentiation and finite differences for derivative computation within PI-MFM. Finally, we demonstrate zero-shot physics-informed fine-tuning to unseen PDE families: starting from a physics-informed pretrained model, adapting using only PDE residuals and initial/boundary conditions, without any labeled solution data, rapidly reduces test errors to around 1% and clearly outperforms physics-only training from scratch. These results show that PI-MFM provides a practical and scalable path toward data-efficient, transferable PDE solvers.

Peiyuan Chen, Zecheng Zhang, Yiping Dong et al.

Visual Question Answering (VQA) is a challenging task that requires systems to provide accurate answers to questions based on image content. Current VQA models struggle with complex questions due to limitations in capturing and integrating multimodal information effectively. To address these challenges, we propose the Rank VQA model, which leverages a ranking-inspired hybrid training strategy to enhance VQA performance. The Rank VQA model integrates high-quality visual features extracted using the Faster R-CNN model and rich semantic text features obtained from a pre-trained BERT model. These features are fused through a sophisticated multimodal fusion technique employing multi-head self-attention mechanisms. Additionally, a ranking learning module is incorporated to optimize the relative ranking of answers, thus improving answer accuracy. The hybrid training strategy combines classification and ranking losses, enhancing the model's generalization ability and robustness across diverse datasets. Experimental results demonstrate the effectiveness of the Rank VQA model. Our model significantly outperforms existing state-of-the-art models on standard VQA datasets, including VQA v2.0 and COCO-QA, in terms of both accuracy and Mean Reciprocal Rank (MRR). The superior performance of Rank VQA is evident in its ability to handle complex questions that require understanding nuanced details and making sophisticated inferences from the image and text. This work highlights the effectiveness of a ranking-based hybrid training strategy in improving VQA performance and lays the groundwork for further research in multimodal learning methods.

Weiyu Ma, Yuqian Fu, Zecheng Zhang et al.

We introduce Attentive VLM Agent (AVA), a multimodal StarCraft II agent that aligns artificial agent perception with the human gameplay experience. Traditional frameworks such as SMAC rely on abstract state representations that diverge significantly from human perception, limiting the ecological validity of agent behavior. Our agent addresses this limitation by incorporating RGB visual inputs and natural language observations that more closely simulate human cognitive processes during gameplay. The AVA architecture consists of three integrated components: (1) a vision-language model enhanced with specialized self-attention mechanisms for strategic unit targeting and battlefield assessment, (2) a retrieval-augmented generation system that leverages domain-specific StarCraft II knowledge to inform tactical decisions, and (3) a dynamic role-based task distribution system that enables coordinated multi-agent behavior. The experimental evaluation in our proposed AVACraft environment, which contains 21 multimodal StarCraft II scenarios, demonstrates that AVA powered by foundation models (specifically Qwen-VL and GPT-4o) can execute complex tactical maneuvers without explicit training, achieving comparable performance to traditional MARL methods that require substantial training iterations. This work establishes a foundation for developing human-aligned StarCraft II agents and advances the broader research agenda of multimodal game AI. Our implementation is available at https://github.com/camel-ai/VLM-Play-StarCraft2.

Zecheng Zhang, Han Zheng, Yue Xu

Evaluating production LLM responses and routing requests across providers in LLM gateways requires fine-grained quality signals and operationally grounded decisions. To address this gap, we present SEAR, a schema-based evaluation and routing system for multi-model, multi-provider LLM gateways. SEAR defines an extensible relational schema covering both LLM evaluation signals (context, intent, response characteristics, issue attribution, and quality scores) and gateway operational metrics (latency, cost, throughput), with cross-table consistency links across around one hundred typed, SQL-queryable columns. To populate the evaluation signals reliably, SEAR proposes self-contained signal instructions, in-schema reasoning, and multi-stage generation that produces database-ready structured outputs. Because signals are derived through LLM reasoning rather than shallow classifiers, SEAR captures complex request semantics, enables human-interpretable routing explanations, and unifies evaluation and routing in a single query layer. Across thousands of production sessions, SEAR achieves strong signal accuracy on human-labeled data and supports practical routing decisions, including large cost reductions with comparable quality.

Huajie Shao, Dachun Sun, Jiahao Wu et al.

GitHub has become a popular social application platform, where a large number of users post their open source projects. In particular, an increasing number of researchers release repositories of source code related to their research papers in order to attract more people to follow their work. Motivated by this trend, we describe a novel item-item cross-platform recommender system, $\textit{paper2repo}$, that recommends relevant repositories on GitHub that match a given paper in an academic search system such as Microsoft Academic. The key challenge is to identify the similarity between an input paper and its related repositories across the two platforms, $\textit{without the benefit of human labeling}$. Towards that end, paper2repo integrates text encoding and constrained graph convolutional networks (GCN) to automatically learn and map the embeddings of papers and repositories into the same space, where proximity offers the basis for recommendation. To make our method more practical in real life systems, labels used for model training are computed automatically from features of user actions on GitHub. In machine learning, such automatic labeling is often called {\em distant supervision\/}. To the authors' knowledge, this is the first distant-supervised cross-platform (paper to repository) matching system. We evaluate the performance of paper2repo on real-world data sets collected from GitHub and Microsoft Academic. Results demonstrate that it outperforms other state of the art recommendation methods.

Erhan Bayraktar, Qi Feng, Zecheng Zhang et al.

We propose a deep neural-operator framework for a general class of probability models. Under global Lipschitz conditions on the operator over the entire Euclidean space-and for a broad class of probabilistic models-we establish a universal approximation theorem with explicit network-size bounds for the proposed architecture. The underlying stochastic processes are required only to satisfy integrability and general tail-probability conditions. We verify these assumptions for both European and American option-pricing problems within the forward-backward SDE (FBSDE) framework, which in turn covers a broad class of operators arising from parabolic PDEs, with or without free boundaries. Finally, we present a numerical example for a basket of American options, demonstrating that the learned model produces optimal stopping boundaries for new strike prices without retraining.

Cangqing Wang, Mingxiu Sui, Dan Sun et al.

This research delves deeply into Meta Reinforcement Learning (Meta RL) through a exploration focusing on defining generalization limits and ensuring convergence. By employing a approach this article introduces an innovative theoretical framework to meticulously assess the effectiveness and performance of Meta RL algorithms. We present an explanation of generalization limits measuring how well these algorithms can adapt to learning tasks while maintaining consistent results. Our analysis delves into the factors that impact the adaptability of Meta RL revealing the relationship, between algorithm design and task complexity. Additionally we establish convergence assurances by proving conditions under which Meta RL strategies are guaranteed to converge towards solutions. We examine the convergence behaviors of Meta RL algorithms across scenarios providing a comprehensive understanding of the driving forces behind their long term performance. This exploration covers both convergence and real time efficiency offering a perspective, on the capabilities of these algorithms.

Xirui Peng, Qiming Xu, Zheng Feng et al.

This paper explores an automatic news generation and fact-checking system based on language processing, aimed at enhancing the efficiency and quality of news production while ensuring the authenticity and reliability of the news content. With the rapid development of Natural Language Processing (NLP) and deep learning technologies, automatic news generation systems are capable of extracting key information from massive data and generating well-structured, fluent news articles. Meanwhile, by integrating fact-checking technology, the system can effectively prevent the spread of false news and improve the accuracy and credibility of news. This study details the key technologies involved in automatic news generation and factchecking, including text generation, information extraction, and the application of knowledge graphs, and validates the effectiveness of these technologies through experiments. Additionally, the paper discusses the future development directions of automatic news generation and fact-checking systems, emphasizing the importance of further integration and innovation of technologies. The results show that with continuous technological optimization and practical application, these systems will play an increasingly important role in the future news industry, providing more efficient and reliable news services.

Jingmin Sun, Yuxuan Liu, Zecheng Zhang et al.

Foundation models, such as large language models, have demonstrated success in addressing various language and image processing tasks. In this work, we introduce a multi-modal foundation model for scientific problems, named PROSE-PDE. Our model, designed for bi-modality to bi-modality learning, is a multi-operator learning approach which can predict future states of spatiotemporal systems while concurrently learning the underlying governing equations of the physical system. Specifically, we focus on multi-operator learning by training distinct one-dimensional time-dependent nonlinear constant coefficient partial differential equations, with potential applications to many physical applications including physics, geology, and biology. More importantly, we provide three extrapolation studies to demonstrate that PROSE-PDE can generalize physical features through the robust training of multiple operators and that the proposed model can extrapolate to predict PDE solutions whose models or data were unseen during the training. Furthermore, we show through systematic numerical experiments that the utilization of the symbolic modality in our model effectively resolves the well-posedness problems with training multiple operators and thus enhances our model's predictive capabilities.

Weihua Hu, Yiwen Yuan, Zecheng Zhang et al. · stanford

We present PyTorch Frame, a PyTorch-based framework for deep learning over multi-modal tabular data. PyTorch Frame makes tabular deep learning easy by providing a PyTorch-based data structure to handle complex tabular data, introducing a model abstraction to enable modular implementation of tabular models, and allowing external foundation models to be incorporated to handle complex columns (e.g., LLMs for text columns). We demonstrate the usefulness of PyTorch Frame by implementing diverse tabular models in a modular way, successfully applying these models to complex multi-modal tabular data, and integrating our framework with PyTorch Geometric, a PyTorch library for Graph Neural Networks (GNNs), to perform end-to-end learning over relational databases.

Christian Moya, Amirhossein Mollaali, Zecheng Zhang et al.

In this paper, we adopt conformal prediction, a distribution-free uncertainty quantification (UQ) framework, to obtain confidence prediction intervals with coverage guarantees for Deep Operator Network (DeepONet) regression. Initially, we enhance the uncertainty quantification frameworks (B-DeepONet and Prob-DeepONet) previously proposed by the authors by using split conformal prediction. By combining conformal prediction with our Prob- and B-DeepONets, we effectively quantify uncertainty by generating rigorous confidence intervals for DeepONet prediction. Additionally, we design a novel Quantile-DeepONet that allows for a more natural use of split conformal prediction. We refer to this distribution-free effective uncertainty quantification framework as split conformal Quantile-DeepONet regression. Finally, we demonstrate the effectiveness of the proposed methods using various ordinary, partial differential equation numerical examples, and multi-fidelity learning.

Matthias Fey, Jinu Sunil, Akihiro Nitta et al.

PyG (PyTorch Geometric) has evolved significantly since its initial release, establishing itself as a leading framework for Graph Neural Networks. In this paper, we present Pyg 2.0 (and its subsequent minor versions), a comprehensive update that introduces substantial improvements in scalability and real-world application capabilities. We detail the framework's enhanced architecture, including support for heterogeneous and temporal graphs, scalable feature/graph stores, and various optimizations, enabling researchers and practitioners to tackle large-scale graph learning problems efficiently. Over the recent years, PyG has been supporting graph learning in a large variety of application areas, which we will summarize, while providing a deep dive into the important areas of relational deep learning and large language modeling.

Zecheng Zhang

The study of operator learning involves the utilization of neural networks to approximate operators. Traditionally, the focus has been on single-operator learning (SOL). However, recent advances have rapidly expanded this to include the approximation of multiple operators using foundation models equipped with millions or billions of trainable parameters, leading to the research of multi-operator learning (MOL). In this paper, we present a novel distributed training approach aimed at enabling a single neural operator with significantly fewer parameters to effectively tackle multi-operator learning challenges, all without incurring additional average costs. Our method is applicable to various neural operators, such as Deep Operator Neural Networks (DON). The core idea is to independently learn the output basis functions for each operator using its dedicated data, while simultaneously centralizing the learning of the input function encoding shared by all operators using the entire dataset. Through a systematic study of five numerical examples, we compare the accuracy and cost of training a single neural operator for each operator independently versus training a MOL model using our proposed method. Our results demonstrate enhanced efficiency and satisfactory accuracy. Moreover, our approach illustrates that some operators with limited data can be more effectively constructed with the aid of data from analogous operators through MOL learning. This highlights another MOL's potential to bolster operator learning.

Yiwen Yuan, Zecheng Zhang, Xinwei He et al.

Recommendation systems predominantly utilize two-tower architectures, which evaluate user-item rankings through the inner product of their respective embeddings. However, one key limitation of two-tower models is that they learn a pair-agnostic representation of users and items. In contrast, pair-wise representations either scale poorly due to their quadratic complexity or are too restrictive on the candidate pairs to rank. To address these issues, we introduce Context-based Graph Neural Networks (ContextGNNs), a novel deep learning architecture for link prediction in recommendation systems. The method employs a pair-wise representation technique for familiar items situated within a user's local subgraph, while leveraging two-tower representations to facilitate the recommendation of exploratory items. A final network then predicts how to fuse both pair-wise and two-tower recommendations into a single ranking of items. We demonstrate that ContextGNN is able to adapt to different data characteristics and outperforms existing methods, both traditional and GNN-based, on a diverse set of practical recommendation tasks, improving performance by 20% on average.

Zecheng Zhang, Christian Moya, Lu Lu et al.

We propose a novel fine-tuning method to achieve multi-operator learning through training a distributed neural operator with diverse function data and then zero-shot fine-tuning the neural network using physics-informed losses for downstream tasks. Operator learning effectively approximates solution operators for PDEs and various PDE-related problems, yet it often struggles to generalize to new tasks. To address this, we investigate fine-tuning a pretrained model, while carefully selecting an initialization that enables rapid adaptation to new tasks with minimal data. Our approach combines distributed learning to integrate data from various operators in pre-training, while physics-informed methods enable zero-shot fine-tuning, minimizing the reliance on downstream data. We investigate standard fine-tuning and Low-Rank Adaptation fine-tuning, applying both to train complex nonlinear target operators that are difficult to learn only using random initialization. Through comprehensive numerical examples, we demonstrate the advantages of our approach, showcasing significant improvements in accuracy. Our findings provide a robust framework for advancing multi-operator learning and highlight the potential of transfer learning techniques in this domain.

Zecheng Zhang, Hao Liu, Wenjing Liao et al.

We propose a Coefficient-to-Basis Network (C2BNet), a novel framework for solving inverse problems within the operator learning paradigm. C2BNet efficiently adapts to different discretizations through fine-tuning, using a pre-trained model to significantly reduce computational cost while maintaining high accuracy. Unlike traditional approaches that require retraining from scratch for new discretizations, our method enables seamless adaptation without sacrificing predictive performance. Furthermore, we establish theoretical approximation and generalization error bounds for C2BNet by exploiting low-dimensional structures in the underlying datasets. Our analysis demonstrates that C2BNet adapts to low-dimensional structures without relying on explicit encoding mechanisms, highlighting its robustness and efficiency. To validate our theoretical findings, we conducted extensive numerical experiments that showcase the superior performance of C2BNet on several inverse problems. The results confirm that C2BNet effectively balances computational efficiency and accuracy, making it a promising tool to solve inverse problems in scientific computing and engineering applications.

Bowen Song, Zecheng Zhang, Zhaoxu Luo et al.

Diffusion models have emerged as powerful tools for generative tasks, producing high-quality outputs across diverse domains. However, how the generated data responds to the initial noise perturbation in diffusion models remains under-explored, which hinders understanding the controllability of the sampling process. In this work, we first observe an interesting phenomenon: the relationship between the change of generation outputs and the scale of initial noise perturbation is highly linear through the diffusion ODE sampling. Then we provide both theoretical and empirical study to justify this linearity property of this input-output (noise-generation data) relationship. Inspired by these new insights, we propose a novel Controllable and Constrained Sampling method (CCS) together with a new controller algorithm for diffusion models to sample with desired statistical properties while preserving good sample quality. We perform extensive experiments to compare our proposed sampling approach with other methods on both sampling controllability and sampled data quality. Results show that our CCS method achieves more precisely controlled sampling while maintaining superior sample quality and diversity.

Adrien Weihs, Jingmin Sun, Zecheng Zhang et al.

While many problems in machine learning focus on learning mappings between finite-dimensional spaces, scientific applications require approximating mappings between function spaces, i.e., operators. We study the problem of learning collections of operators and provide both theoretical and empirical advances. We distinguish between two regimes: (i) multiple operator learning, where a single network represents a continuum of operators parameterized by a parametric function, and (ii) learning several distinct single operators, where each operator is learned independently. For the multiple operator case, we introduce two new architectures, $\mathrm{MNO}$ and $\mathrm{MONet}$, and establish universal approximation results in three settings: continuous, integrable, or Lipschitz operators. For the latter, we further derive explicit scaling laws that quantify how the network size must grow to achieve a target approximation accuracy. For learning several single operators, we develop a framework for balancing architectural complexity across subnetworks and show how approximation order determines computational efficiency. Empirical experiments on parametric PDE benchmarks confirm the strong expressive power and efficiency of the proposed architectures. Overall, this work establishes a unified theoretical and practical foundation for scalable neural operator learning across multiple operators.

Qianhui Wan, Zecheng Zhang, Liheng Jiang et al.

Image anomaly detection is a popular research direction, with many methods emerging in recent years due to rapid advancements in computing. The use of artificial intelligence for image anomaly detection has been widely studied. By analyzing images of athlete posture and movement, it is possible to predict injury status and suggest necessary adjustments. Most existing methods rely on convolutional networks to extract information from irrelevant pixel data, limiting model accuracy. This paper introduces a network combining Residual Network (ResNet) and Bidirectional Gated Recurrent Unit (BiGRU), which can predict potential injury types and provide early warnings by analyzing changes in muscle and bone poses from video images. To address the high complexity of this network, the Sparrow search algorithm was used for optimization. Experiments conducted on four datasets demonstrated that our model has the smallest error in image anomaly detection compared to other models, showing strong adaptability. This provides a new approach for anomaly detection and predictive analysis in images, contributing to the sustainable development of human health and performance.

Guang Lin, Christian Moya, Zecheng Zhang

The Deep Operator Networks~(DeepONet) is a fundamentally different class of neural networks that we train to approximate nonlinear operators, including the solution operator of parametric partial differential equations (PDE). DeepONets have shown remarkable approximation and generalization capabilities even when trained with relatively small datasets. However, the performance of DeepONets deteriorates when the training data is polluted with noise, a scenario that occurs very often in practice. To enable DeepONets training with noisy data, we propose using the Bayesian framework of replica-exchange Langevin diffusion. Such a framework uses two particles, one for exploring and another for exploiting the loss function landscape of DeepONets. We show that the proposed framework's exploration and exploitation capabilities enable (1) improved training convergence for DeepONets in noisy scenarios and (2) attaching an uncertainty estimate for the predicted solutions of parametric PDEs. In addition, we show that replica-exchange Langeving Diffusion (remarkably) also improves the DeepONet's mean prediction accuracy in noisy scenarios compared with vanilla DeepONets trained with state-of-the-art gradient-based optimization algorithms (e.g. Adam). To reduce the potentially high computational cost of replica, in this work, we propose an accelerated training framework for replica-exchange Langevin diffusion that exploits the neural network architecture of DeepONets to reduce its computational cost up to 25% without compromising the proposed framework's performance. Finally, we illustrate the effectiveness of the proposed Bayesian framework using a series of experiments on four parametric PDE problems.

Yuxin Xiao, Zecheng Zhang, Yuning Mao et al.

Stepping from sentence-level to document-level, the research on relation extraction (RE) confronts increasing text length and more complicated entity interactions. Consequently, it is more challenging to encode the key information sources--relevant contexts and entity types. However, existing methods only implicitly learn to model these critical information sources while being trained for RE. As a result, they suffer the problems of ineffective supervision and uninterpretable model predictions. In contrast, we propose to explicitly teach the model to capture relevant contexts and entity types by supervising and augmenting intermediate steps (SAIS) for RE. Based on a broad spectrum of carefully designed tasks, our proposed SAIS method not only extracts relations of better quality due to more effective supervision, but also retrieves the corresponding supporting evidence more accurately so as to enhance interpretability. By assessing model uncertainty, SAIS further boosts the performance via evidence-based data augmentation and ensemble inference while reducing the computational cost. Eventually, SAIS delivers state-of-the-art RE results on three benchmarks (DocRED, CDR, and GDA) and outperforms the runner-up by 5.04% relatively in F1 score in evidence retrieval on DocRED.

Eric Chung, Yalchin Efendiev, Wing Tat Leung et al.

In this work, we propose a multi-agent actor-critic reinforcement learning (RL) algorithm to accelerate the multi-level Monte Carlo Markov Chain (MCMC) sampling algorithms. The policies (actors) of the agents are used to generate the proposal in the MCMC steps; and the critic, which is centralized, is in charge of estimating the long term reward. We verify our proposed algorithm by solving an inverse problem with multiple scales. There are several difficulties in the implementation of this problem by using traditional MCMC sampling. Firstly, the computation of the posterior distribution involves evaluating the forward solver, which is very time consuming for a problem with heterogeneous. We hence propose to use the multi-level algorithm. More precisely, we use the generalized multiscale finite element method (GMsFEM) as the forward solver in evaluating a posterior distribution in the multi-level rejection procedure. Secondly, it is hard to find a function which can generate samplings which are meaningful. To solve this issue, we learn an RL policy as the proposal generator. Our experiments show that the proposed method significantly improves the sampling process