Weinan E

Lin Lin, Jianfeng Lu, Roberto Car et al.

We propose a multipole representation of the Fermi-Dirac function and the Fermi operator, and use this representation to develop algorithms for electronic structure analysis of metallic systems. The new algorithm is quite simple and efficient. Its computational cost scales logarithmically with $βΔ\eps$ where $β$ is the inverse temperature, and $Δ\eps$ is the width of the spectrum of the discretized Hamiltonian matrix.

Weinan E, Martin Hutzenthaler, Arnulf Jentzen et al.

We introduce a new family of numerical algorithms for approximating solutions of general high-dimensional semilinear parabolic partial differential equations at single space-time points. The algorithm is obtained through a delicate combination of the Feynman-Kac and the Bismut-Elworthy-Li formulas, and an approximate decomposition of the Picard fixed-point iteration with multilevel accuracy. The algorithm has been tested on a variety of semilinear partial differential equations that arise in physics and finance, with very satisfactory results. Analytical tools needed for the analysis of such algorithms, including a semilinear Feynman-Kac formula, a new class of semi-norms and their recursive inequalities, are also introduced. They allow us to prove for semilinear heat equations with gradient-independent nonlinearity that the computational complexity of the proposed algorithm is bounded by $O(d\,\varepsilon^{-(4+δ)})$ for any $δ\in (0,\infty)$ under suitable assumptions, where $d\in \mathbb{N}$ is the dimensionality of the problem and $\varepsilon\in(0,\infty)$ is the prescribed accuracy.

Xinyu Zhu, Yuzhu Cai, Zexi Liu et al.

The convergence of large language models and agents is catalyzing a new era of scientific discovery: Agentic Science. While the scientific method is inherently iterative, existing agent frameworks are predominantly static, narrowly scoped, and lack the capacity to learn from trial and error. To bridge this gap, we present EvoMaster, a foundational evolving agent framework engineered specifically for Agentic Science at Scale. Driven by the core principle of continuous self-evolution, EvoMaster empowers agents to iteratively refine hypotheses, self-critique, and progressively accumulate knowledge across experimental cycles, faithfully mirroring human scientific inquiry. Crucially, as a domain-agnostic base harness, EvoMaster is exceptionally easy to scale up -- enabling developers to build and deploy highly capable, self-evolving scientific agents for arbitrary disciplines in approximately 100 lines of code. Built upon EvoMaster, we incubated the SciMaster ecosystem across domains such as machine learning, physics, and general science. Evaluations on four authoritative benchmarks (Humanity's Last Exam, MLE-Bench Lite, BrowseComp, and FrontierScience) demonstrate that EvoMaster achieves state-of-the-art scores of 41.1%, 75.8%, 73.3%, and 53.3%, respectively. It comprehensively outperforms the general-purpose baseline OpenClaw with relative improvements ranging from +159% to +316%, robustly validating its efficacy and generality as the premier foundational framework for the next generation of autonomous scientific discovery. EvoMaster is available at https://github.com/sjtu-sai-agents/EvoMaster.

Lin Lin, Sihong Shao, Weinan E

We present for the first time an efficient iterative method to directly solve the four-component Dirac-Kohn-Sham (DKS) density functional theory. Due to the existence of the negative energy continuum in the DKS operator, the existing iterative techniques for solving the Kohn-Sham systems cannot be efficiently applied to solve the DKS systems. The key component of our method is a novel filtering step (F) which acts as a preconditioner in the framework of the locally optimal block preconditioned conjugate gradient (LOBPCG) method. The resulting method, dubbed the LOBPCG-F method, is able to compute the desired eigenvalues and eigenvectors in the positive energy band without computing any state in the negative energy band. The LOBPCG-F method introduces mild extra cost compared to the standard LOBPCG method and can be easily implemented. We demonstrate our method in the pseudopotential framework with a planewave basis set which naturally satisfies the kinetic balance prescription. Numerical results for Pt$_{2}$, Au$_{2}$, TlF, and Bi$_{2}$Se$_{3}$ indicate that the LOBPCG-F method is a robust and efficient method for investigating the relativistic effect in systems containing heavy elements.

Weinan E, Eric Vanden-Eijnden

The "equation-free'' approach has been proposed in recent years as a general framework for developing multiscale methods to efficiently capture the macroscale behavior of a system using only the microscale models. In this paper, we take a close look at some of the algorithms proposed under the "equation-free'' umbrella, the projective integrators and the patch dynamics. We discuss some very simple examples in the context of the "equation-free'' approach. These examples seem to indicate that while its general philosophy is quite attractive and indeed similar to many other approaches in concurrent multiscale modeling, there are severe limitations to the specific implementation proposed by the equation-free approach.

Zeping Min, Qian Ge, Zhong Li et al.

We propose a unified framework for low resource automatic speech recognition tasks named meta audio concatenation (MAC). It is easy to implement and can be carried out in extremely low resource environments. Mathematically, we give a clear description of MAC framework from the perspective of bayesian sampling. In this framework, we leverage a novel concatenative synthesis text-to-speech system to boost the low resource ASR task. By the concatenative synthesis text-to-speech system, we can integrate language pronunciation rules and adjust the TTS process. Furthermore, we propose a broad notion of meta audio set to meet the modeling needs of different languages and different scenes when using the system. Extensive experiments have demonstrated the great effectiveness of MAC on low resource ASR tasks. For CTC greedy search, CTC prefix, attention, and attention rescoring decode mode in Cantonese ASR task, Taiwanese ASR task, and Japanese ASR task the MAC method can reduce the CER by more than 15\%. Furthermore, in the ASR task, MAC beats wav2vec2 (with fine-tuning) on common voice datasets of Cantonese and gets really competitive results on common voice datasets of Taiwanese and Japanese. Among them, it is worth mentioning that we achieve a \textbf{10.9\%} character error rate (CER) on the common voice Cantonese ASR task, bringing about \textbf{30\%} relative improvement compared to the wav2vec2 (with fine-tuning).

Hongkang Yang, Zehao Lin, Wenjin Wang et al.

The training and inference of large language models (LLMs) are together a costly process that transports knowledge from raw data to meaningful computation. Inspired by the memory hierarchy of the human brain, we reduce this cost by equipping LLMs with explicit memory, a memory format cheaper than model parameters and text retrieval-augmented generation (RAG). Conceptually, with most of its knowledge externalized to explicit memories, the LLM can enjoy a smaller parameter size, training cost, and inference cost, all proportional to the amount of remaining "abstract knowledge". As a preliminary proof of concept, we train from scratch a 2.4B LLM, which achieves better performance than much larger LLMs as well as RAG models, and maintains higher decoding speed than RAG. The model is named $\text{Memory}^3$, since explicit memory is the third form of memory in LLMs after implicit memory (model parameters) and working memory (context key-values). We introduce a memory circuitry theory to support the externalization of knowledge, and present novel techniques including a memory sparsification mechanism that makes storage tractable and a two-stage pretraining scheme that facilitates memory formation.

Zichen Wen, Boxue Yang, Shuang Chen et al.

We present Innovator-VL, a scientific multimodal large language model designed to advance understanding and reasoning across diverse scientific domains while maintaining excellent performance on general vision tasks. Contrary to the trend of relying on massive domain-specific pretraining and opaque pipelines, our work demonstrates that principled training design and transparent methodology can yield strong scientific intelligence with substantially reduced data requirements. (i) First, we provide a fully transparent, end-to-end reproducible training pipeline, covering data collection, cleaning, preprocessing, supervised fine-tuning, reinforcement learning, and evaluation, along with detailed optimization recipes. This facilitates systematic extension by the community. (ii) Second, Innovator-VL exhibits remarkable data efficiency, achieving competitive performance on various scientific tasks using fewer than five million curated samples without large-scale pretraining. These results highlight that effective reasoning can be achieved through principled data selection rather than indiscriminate scaling. (iii) Third, Innovator-VL demonstrates strong generalization, achieving competitive performance on general vision, multimodal reasoning, and scientific benchmarks. This indicates that scientific alignment can be integrated into a unified model without compromising general-purpose capabilities. Our practices suggest that efficient, reproducible, and high-performing scientific multimodal models can be built even without large-scale data, providing a practical foundation for future research.

Xinyu Zhu, Yuzhu Cai, Zexi Liu et al.

The advancement of artificial intelligence toward agentic science is currently bottlenecked by the challenge of ultra-long-horizon autonomy, the ability to sustain strategic coherence and iterative correction over experimental cycles spanning days or weeks. While Large Language Models (LLMs) have demonstrated prowess in short-horizon reasoning, they are easily overwhelmed by execution details in the high-dimensional, delayed-feedback environments of real-world research, failing to consolidate sparse feedback into coherent long-term guidance. Here, we present ML-Master 2.0, an autonomous agent that masters ultra-long-horizon machine learning engineering (MLE) which is a representative microcosm of scientific discovery. By reframing context management as a process of cognitive accumulation, our approach introduces Hierarchical Cognitive Caching (HCC), a multi-tiered architecture inspired by computer systems that enables the structural differentiation of experience over time. By dynamically distilling transient execution traces into stable knowledge and cross-task wisdom, HCC allows agents to decouple immediate execution from long-term experimental strategy, effectively overcoming the scaling limits of static context windows. In evaluations on OpenAI's MLE-Bench under 24-hour budgets, ML-Master 2.0 achieves a state-of-the-art medal rate of 56.44%. Our findings demonstrate that ultra-long-horizon autonomy provides a scalable blueprint for AI capable of autonomous exploration beyond human-precedent complexities.

Hao Liang, Zhengyang Zhao, Meiyi Qiang et al.

Data-centric training has emerged as a promising direction for improving large language models (LLMs) by optimizing not only model parameters but also the selection, composition, and weighting of training data during optimization. However, existing approaches to data selection, data mixture optimization, and data reweighting are often developed in isolated codebases with inconsistent interfaces, hindering reproducibility, fair comparison, and practical integration. In this paper, we present DataFlex, a unified data-centric dynamic training framework built upon LLaMA-Factory. DataFlex supports three major paradigms of dynamic data optimization: sample selection, domain mixture adjustment, and sample reweighting, while remaining fully compatible with the original training workflow. It provides extensible trainer abstractions and modular components, enabling a drop-in replacement for standard LLM training, and unifies key model-dependent operations such as embedding extraction, inference, and gradient computation, with support for large-scale settings including DeepSpeed ZeRO-3. We conduct comprehensive experiments across multiple data-centric methods. Dynamic data selection consistently outperforms static full-data training on MMLU across both Mistral-7B and Llama-3.2-3B. For data mixture, DoReMi and ODM improve both MMLU accuracy and corpus-level perplexity over default proportions when pretraining Qwen2.5-1.5B on SlimPajama at 6B and 30B token scales. DataFlex also achieves consistent runtime improvements over original implementations. These results demonstrate that DataFlex provides an effective, efficient, and reproducible infrastructure for data-centric dynamic training of LLMs.

Hao Liang, Xiaochen Ma, Zhou Liu et al.

The rapidly growing demand for high-quality data in Large Language Models (LLMs) has intensified the need for scalable, reliable, and semantically rich data preparation pipelines. However, current practices remain dominated by ad-hoc scripts and loosely specified workflows, which lack principled abstractions, hinder reproducibility, and offer limited support for model-in-the-loop data generation. To address these challenges, we present DataFlow, a unified and extensible LLM-driven data preparation framework. DataFlow is designed with system-level abstractions that enable modular, reusable, and composable data transformations, and provides a PyTorch-style pipeline construction API for building debuggable and optimizable dataflows. The framework consists of nearly 200 reusable operators and six domain-general pipelines spanning text, mathematical reasoning, code, Text-to-SQL, agentic RAG, and large-scale knowledge extraction. To further improve usability, we introduce DataFlow-Agent, which automatically translates natural-language specifications into executable pipelines via operator synthesis, pipeline planning, and iterative verification. Across six representative use cases, DataFlow consistently improves downstream LLM performance. Our math, code, and text pipelines outperform curated human datasets and specialized synthetic baselines, achieving up to +3\% execution accuracy in Text-to-SQL over SynSQL, +7\% average improvements on code benchmarks, and 1--3 point gains on MATH, GSM8K, and AIME. Moreover, a unified 10K-sample dataset produced by DataFlow enables base models to surpass counterparts trained on 1M Infinity-Instruct data. These results demonstrate that DataFlow provides a practical and high-performance substrate for reliable, reproducible, and scalable LLM data preparation, and establishes a system-level foundation for future data-centric AI development.

Tingjia Miao, Wenkai Jin, Muhua Zhang et al.

The paradigm of agentic science requires AI systems to conduct robust reasoning and engage in long-horizon, autonomous exploration. However, current scientific benchmarks remain confined to domain knowledge comprehension and complex reasoning, failing to evaluate the exploratory nature and procedural complexity of real-world research. In this work, we present research-oriented evaluations in theoretical and computational physics, a natural testbed with comprehensive domain knowledge, complex reasoning, and verifiable end-to-end workflows without reliance on experiments. Here we introduce PRL-Bench (Physics Research by LLMs), a benchmark designed to systematically map the capability boundaries of LLMs in executing end-to-end physics research. Constructed from 100 curated papers from the latest issues of Physical Review Letters since August 2025 and validated by domain experts, PRL-Bench covers five major theory- and computation-intensive subfields of modern physics: astrophysics, condensed matter physics, high-energy physics, quantum information, and statistical physics. Each task in the benchmark is designed to replicate the core properties of authentic scientific research, including exploration-oriented formulation, long-horizon workflows, and objective verifiability, thereby reconstructing the essential reasoning processes and research workflows of real physics research. Evaluation across frontier models shows that performance remains limited, with the best overall score below 50, revealing a pronounced gap between current LLM capabilities and the demands of real scientific research. PRL-Bench serves a reliable testbed for accessing next generation AI scientists advancing AI systems toward autonomous scientific discovery.

Yichen He, Guanhua Huang, Peiyuan Feng et al.

We introduce PaSa, an advanced Paper Search agent powered by large language models. PaSa can autonomously make a series of decisions, including invoking search tools, reading papers, and selecting relevant references, to ultimately obtain comprehensive and accurate results for complex scholar queries. We optimize PaSa using reinforcement learning with a synthetic dataset, AutoScholarQuery, which includes 35k fine-grained academic queries and corresponding papers sourced from top-tier AI conference publications. Additionally, we develop RealScholarQuery, a benchmark collecting real-world academic queries to assess PaSa performance in more realistic scenarios. Despite being trained on synthetic data, PaSa significantly outperforms existing baselines on RealScholarQuery, including Google, Google Scholar, Google with GPT-4o for paraphrased queries, ChatGPT (search-enabled GPT-4o), GPT-o1, and PaSa-GPT-4o (PaSa implemented by prompting GPT-4o). Notably, PaSa-7B surpasses the best Google-based baseline, Google with GPT-4o, by 37.78% in recall@20 and 39.90% in recall@50, and exceeds PaSa-GPT-4o by 30.36% in recall and 4.25% in precision. Model, datasets, and code are available at https://github.com/bytedance/pasa.

Yuheng Wang, Runde Yang, Lin Wu et al.

The scalability of high-quality online education is hindered by the high costs and slow cycles of manual content creation. Despite advancements in video generation, current approaches often fail to ensure pedagogical structure and precise control due to their pixel-level, black-box nature. In this paper, we propose Generative Teaching, a novel paradigm shifting educators from manual creators to high-level directors who focus on pedagogical intents while agents handle the execution. To realize this vision, we introduce TeachMaster, a multi-agent framework that leverages code as an intermediate semantic medium. Unlike traditional video generation methods, TeachMaster orchestrates a collaborative team of agents, spanning planning, design, and rendering, to automate the production of interpretable, editable, and curriculum-ready educational videos. Experiments validate that TeachMaster significantly boosts production efficiency without compromising structural coherence or visual fidelity, slashing production costs to only 0.3% of traditional online course videos and providing a robust solution for scalable education.

Jingyi Chai, Shuo Tang, Rui Ye et al.

The rapid advancements of AI agents have ignited the long-held ambition of leveraging them to accelerate scientific discovery. Achieving this goal requires a deep understanding of the frontiers of human knowledge. As such, Humanity's Last Exam (HLE) provides an exceptionally challenging touchstone for evaluating scientific AI agents. In this work, we aim to construct the foundational architecture for general-purpose agents and validate the capabilities through leading performance on HLE. To achieve this, we introduce X-Master, a tool-augmented reasoning agent designed to emulate human researchers by interacting flexibly with external tools during its reasoning process. This agent, guided by the conceptualization of code as an interaction language, can flexibly leverage built-in Python libraries and our customized tools to augment the reasoning. We further scale its capabilities through X-Masters, a scattered-and-stacked agentic workflow that systematically enhances breadth and depth of reasoning. Our open-source solution, X-Masters, sets a new state-of-the-art record on HLE with a score of 32.1%, surpassing OpenAI's and Google's Deep Research (26.6% and 26.9%) and becoming the first to exceed the 30% threshold. This work allows us to gain a deeper understanding of complex task-solving and accumulates valuable experience that can inform future advancements, guiding subsequent model training.

Linfeng Zhang, Siheng Chen, Yuzhu Cai et al.

AI agents are emerging as a practical way to run multi-step scientific workflows that interleave reasoning with tool use and verification, pointing to a shift from isolated AI-assisted steps toward \emph{agentic science at scale}. This shift is increasingly feasible, as scientific tools and models can be invoked through stable interfaces and verified with recorded execution traces, and increasingly necessary, as AI accelerates scientific output and stresses the peer-review and publication pipeline, raising the bar for traceability and credible evaluation. However, scaling agentic science remains difficult: workflows are hard to observe and reproduce; many tools and laboratory systems are not agent-ready; execution is hard to trace and govern; and prototype AI Scientist systems are often bespoke, limiting reuse and systematic improvement from real workflow signals. We argue that scaling agentic science requires an infrastructure-and-ecosystem approach, instantiated in Bohrium+SciMaster. Bohrium acts as a managed, traceable hub for AI4S assets -- akin to a HuggingFace of AI for Science -- that turns diverse scientific data, software, compute, and laboratory systems into agent-ready capabilities. SciMaster orchestrates these capabilities into long-horizon scientific workflows, on which scientific agents can be composed and executed. Between infrastructure and orchestration, a \emph{scientific intelligence substrate} organizes reusable models, knowledge, and components into executable building blocks for workflow reasoning and action, enabling composition, auditability, and improvement through use. We demonstrate this stack with eleven representative master agents in real workflows, achieving orders-of-magnitude reductions in end-to-end scientific cycle time and generating execution-grounded signals from real workloads at multi-million scale.

Boshen Zeng, Sian Chen, Xinxin Liu et al.

Advancements in lithium battery technology heavily rely on the design and engineering of electrolytes. However, current schemes for molecular design and recipe optimization of electrolytes lack an effective computational-experimental closed loop and often fall short in accurately predicting diverse electrolyte formulation properties. In this work, we introduce Uni-ELF, a novel multi-level representation learning framework to advance electrolyte design. Our approach involves two-stage pretraining: reconstructing three-dimensional molecular structures at the molecular level using the Uni-Mol model, and predicting statistical structural properties (e.g., radial distribution functions) from molecular dynamics simulations at the mixture level. Through this comprehensive pretraining, Uni-ELF is able to capture intricate molecular and mixture-level information, which significantly enhances its predictive capability. As a result, Uni-ELF substantially outperforms state-of-the-art methods in predicting both molecular properties (e.g., melting point, boiling point, synthesizability) and formulation properties (e.g., conductivity, Coulombic efficiency). Moreover, Uni-ELF can be seamlessly integrated into an automatic experimental design workflow. We believe this innovative framework will pave the way for automated AI-based electrolyte design and engineering.

Yongqi Jin, Yecheng Wang, Jun-jie Wang et al.

Accurate prediction of nuclear magnetic resonance (NMR) chemical shifts is fundamental to spectral analysis and molecular structure elucidation, yet existing machine learning methods rely on limited, labor-intensive atom-assigned datasets. We propose a semi-supervised framework that learns NMR chemical shifts from millions of literature-extracted spectra without explicit atom-level assignments, integrating a small amount of labeled data with large-scale unassigned spectra. We formulate chemical shift prediction from literature spectra as a permutation-invariant set supervision problem, and show that under commonly satisfied conditions on the loss function, optimal bipartite matching reduces to a sorting-based loss, enabling stable large-scale semi-supervised training beyond traditional curated datasets. Our models achieve substantially improved accuracy and robustness over state-of-the-art methods and exhibit stronger generalization on significantly larger and more diverse molecular datasets. Moreover, by incorporating solvent information at scale, our approach captures systematic solvent effects across common NMR solvents for the first time. Overall, our results demonstrate that large-scale unlabeled spectra mined from the literature can serve as a practical and effective data source for training NMR shift models, suggesting a broader role of literature-derived, weakly structured data in data-centric AI for science.

Tingjia Miao, Jiawen Dai, Jingkun Liu et al.

Advances in LLMs have produced agents with knowledge and operational capabilities comparable to human scientists, suggesting potential to assist, accelerate, and automate research. However, existing studies mainly evaluate such systems on well-defined benchmarks or general tasks like literature retrieval, limiting their end-to-end problem-solving ability in open scientific scenarios. This is particularly true in physics, which is abstract, mathematically intensive, and requires integrating analytical reasoning with code-based computation. To address this, we propose PhysMaster, an LLM-based agent functioning as an autonomous theoretical and computational physicist. PhysMaster couples absract reasoning with numerical computation and leverages LANDAU, the Layered Academic Data Universe, which preserves retrieved literature, curated prior knowledge, and validated methodological traces, enhancing decision reliability and stability. It also employs an adaptive exploration strategy balancing efficiency and open-ended exploration, enabling robust performance in ultra-long-horizon tasks. We evaluate PhysMaster on problems from high-energy theory, condensed matter theory to astrophysics, including: (i) acceleration, compressing labor-intensive research from months to hours; (ii) automation, autonomously executing hypothesis-driven loops ; and (iii) autonomous discovery, independently exploring open problems.

Hao Liang, Zhengyang Zhao, Zhaoyang Han et al.

Large language models (LLMs) have demonstrated remarkable performance across a wide range of tasks and domains, with data playing a central role in enabling these advances. Despite this success, the preparation and effective utilization of the massive datasets required for LLM training remain major bottlenecks. In current practice, LLM training data is often constructed using ad hoc scripts, and there is still a lack of mature, agent-based data preparation systems that can automatically construct robust and reusable data workflows, thereby freeing data scientists from repetitive and error-prone engineering efforts. Moreover, once collected, datasets are often consumed largely in their entirety during training, without systematic mechanisms for data selection, mixture optimization, or reweighting. To address these limitations, we advocate two complementary research directions. First, we propose building a robust, agent-based automatic data preparation system that supports automated workflow construction and scalable data management. Second, we argue for a unified data-model interaction training system in which data is dynamically selected, mixed, and reweighted throughout the training process, enabling more efficient, adaptive, and performance-aware data utilization. Finally, we discuss the remaining challenges and outline promising directions for future research and system development.

Yucheng Liao, Han Wen, Weinan E et al.

Data-independent acquisition mass spectrometry (DIA-MS) has established itself as a cornerstone of proteomic profiling and large-scale systems biology, offering unparalleled depth and reproducibility. Current DIA analysis frameworks, however, require semi-supervised training within each run for peptide-spectrum match (PSM) re-scoring. This approach is prone to overfitting and lacks generalizability across diverse species and experimental conditions. Here, we present DIA-CLIP, a pre-trained model shifting the DIA analysis paradigm from semi-supervised training to universal cross-modal representation learning. By integrating dual-encoder contrastive learning framework with encoder-decoder architecture, DIA-CLIP establishes a unified cross-modal representation for peptides and corresponding spectral features, achieving high-precision, zero-shot PSM inference. Extensive evaluations across diverse benchmarks demonstrate that DIA-CLIP consistently outperforms state-of-the-art tools, yielding up to a 45% increase in protein identification while achieving a 12% reduction in entrapment identifications. Moreover, DIA-CLIP holds immense potential for diverse practical applications, such as single-cell and spatial proteomics, where its enhanced identification depth facilitates the discovery of novel biomarkers and the elucidates of intricate cellular mechanisms.

Weinan E, Yajun Zhou

We characterize the meaning of words with language-independent numerical fingerprints, through a mathematical analysis of recurring patterns in texts. Approximating texts by Markov processes on a long-range time scale, we are able to extract topics, discover synonyms, and sketch semantic fields from a particular document of moderate length, without consulting external knowledge-base or thesaurus. Our Markov semantic model allows us to represent each topical concept by a low-dimensional vector, interpretable as algebraic invariants in succinct statistical operations on the document, targeting local environments of individual words. These language-independent semantic representations enable a robot reader to both understand short texts in a given language (automated question-answering) and match medium-length texts across different languages (automated word translation). Our semantic fingerprints quantify local meaning of words in 14 representative languages across 5 major language families, suggesting a universal and cost-effective mechanism by which human languages are processed at the semantic level. Our protocols and source codes are publicly available on https://github.com/yajun-zhou/linguae-naturalis-principia-mathematica

Yu Li, Yuan Huang, Tao Wang et al.

Most scientific materials compress reasoning, presenting conclusions while omitting the derivational chains that justify them. This compression hinders verification by lacking explicit, step-wise justifications and inhibits cross-domain links by collapsing the very pathways that establish the logical and causal connections between concepts. We introduce a scalable framework that decompresses scientific reasoning, constructing a verifiable Long Chain-of-Thought (LCoT) knowledge base and projecting it into an emergent encyclopedia, SciencePedia. Our pipeline operationalizes an endpoint-driven, reductionist strategy: a Socratic agent, guided by a curriculum of around 200 courses, generates approximately 3 million first-principles questions. To ensure high fidelity, multiple independent solver models generate LCoTs, which are then rigorously filtered by prompt sanitization and cross-model answer consensus, retaining only those with verifiable endpoints. This verified corpus powers the Brainstorm Search Engine, which performs inverse knowledge search -- retrieving diverse, first-principles derivations that culminate in a target concept. This engine, in turn, feeds the Plato synthesizer, which narrates these verified chains into coherent articles. The initial SciencePedia comprises approximately 200,000 fine-grained entries spanning mathematics, physics, chemistry, biology, engineering, and computation. In evaluations across six disciplines, Plato-synthesized articles (conditioned on retrieved LCoTs) exhibit substantially higher knowledge-point density and significantly lower factual error rates than an equally-prompted baseline without retrieval (as judged by an external LLM). Built on this verifiable LCoT knowledge base, this reasoning-centric approach enables trustworthy, cross-domain scientific synthesis at scale and establishes the foundation for an ever-expanding encyclopedia.

Mingze Wang, Weinan E

We conduct a systematic study of the approximation properties of Transformer for sequence modeling with long, sparse and complicated memory. We investigate the mechanisms through which different components of Transformer, such as the dot-product self-attention, positional encoding and feed-forward layer, affect its expressive power, and we study their combined effects through establishing explicit approximation rates. Our study reveals the roles of critical parameters in the Transformer, such as the number of layers and the number of attention heads. These theoretical insights are validated experimentally and offer natural suggestions for alternative architectures.

Zexi Liu, Yuzhu Cai, Xinyu Zhu et al.

As AI capabilities advance toward and potentially beyond human-level performance, a natural transition emerges where AI-driven development becomes more efficient than human-centric approaches. A promising pathway toward this transition lies in AI-for-AI (AI4AI), which leverages AI techniques to automate and optimize the design, training, and deployment of AI systems themselves. While LLM-based agents have shown the potential to realize AI4AI, they are often unable to fully leverage the experience accumulated by agents during the exploration of solutions in the reasoning process, leading to inefficiencies and suboptimal performance. To address this limitation, we propose ML-Master, a novel AI4AI agent that seamlessly integrates exploration and reasoning by employing a selectively scoped memory mechanism. This approach allows ML-Master to efficiently combine diverse insights from parallel solution trajectories with analytical reasoning, guiding further exploration without overwhelming the agent with excessive context. We evaluate ML-Master on the MLE-Bench, where it achieves a 29.3% average medal rate, significantly surpassing existing methods, particularly in medium-complexity tasks, while accomplishing this superior performance within a strict 12-hour time constraint-half the 24-hour limit used by previous baselines. These results demonstrate ML-Master's potential as a powerful tool for advancing AI4AI.

Yaorui Shi, Sihang Li, Taiyan Zhang et al.

Automated drug discovery offers significant potential for accelerating the development of novel therapeutics by substituting labor-intensive human workflows with machine-driven processes. However, molecules generated by artificial intelligence may unintentionally infringe on existing patents, posing legal and financial risks that impede the full automation of drug discovery pipelines. This paper introduces PatentFinder, a novel multi-agent and tool-enhanced intelligence system that can accurately and comprehensively evaluate small molecules for patent infringement. PatentFinder features five specialized agents that collaboratively analyze patent claims and molecular structures with heuristic and model-based tools, generating interpretable infringement reports. To support systematic evaluation, we curate MolPatent-240, a benchmark dataset tailored for patent infringement assessment algorithms. On this benchmark, PatentFinder outperforms baseline methods that rely solely on large language models or specialized chemical tools, achieving a 13.8% improvement in F1-score and a 12% increase in accuracy. Additionally, PatentFinder autonomously generates detailed and interpretable patent infringement reports, showcasing enhanced accuracy and improved interpretability. The high accuracy and interpretability of PatentFinder make it a valuable and reliable tool for automating patent infringement assessments, offering a practical solution for integrating patent protection analysis into the drug discovery pipeline.

Zhengren Wang, Jiayang Yu, Dongsheng Ma et al.

Domain-specific intelligence demands specialized knowledge and sophisticated reasoning for problem-solving, posing significant challenges for large language models (LLMs) that struggle with knowledge hallucination and inadequate reasoning capabilities under constrained parameter budgets. Inspired by Bloom's Taxonomy in educational theory, we propose Retrieval-Augmented Reasoning Modeling (RARE), a novel paradigm that decouples knowledge storage from reasoning optimization. RARE externalizes domain knowledge to retrievable sources and internalizes domain-specific reasoning patterns during training. Specifically, by injecting retrieved knowledge into training prompts with masked losses, RARE transforms learning objectives from rote memorization to contextualized reasoning. It enables models to bypass parameter-intensive memorization and prioritize the development of higher-order cognitive processes. Extensive experiments demonstrate that lightweight RARE-trained models (e.g., Llama-3.1-8B) could achieve state-of-the-art performance, surpassing retrieval-augmented GPT-4 and DeepSeek-R1 up to approximately 20\% accuracy. RARE establishes a paradigm shift where maintainable external knowledge bases synergize with compact, reasoning-optimized models, collectively driving more scalable domain-specific intelligence.

Pinchen Xie, Yunrui Qiu, Weinan E

A data-driven ab initio generalized Langevin equation (AIGLE) approach is developed to learn and simulate high-dimensional, heterogeneous, coarse-grained conformational dynamics. Constrained by the fluctuation-dissipation theorem, the approach can build coarse-grained models in dynamical consistency with all-atom molecular dynamics. We also propose practical criteria for AIGLE to enforce long-term dynamical consistency. Case studies of a toy polymer, with 20 coarse-grained sites, and the alanine dipeptide, with two dihedral angles, elucidate why one should adopt AIGLE or its Markovian limit for modeling coarse-grained conformational dynamics in practice.

Jiequn Han, Arnulf Jentzen, Weinan E

High-dimensional partial differential equations (PDEs) pose significant challenges for numerical computation due to the curse of dimensionality, which limits the applicability of traditional mesh-based methods. Since 2017, the Deep BSDE method has introduced deep learning techniques that enable the effective solution of nonlinear PDEs in very high dimensions. This innovation has sparked considerable interest in using neural networks for high-dimensional PDEs, making it an active area of research. In this short review, we briefly sketch the Deep BSDE method, its subsequent developments, and future directions for the field.

Jinbo Wang, Mingze Wang, Zhanpeng Zhou et al.

Transformers consist of diverse building blocks, such as embedding layers, normalization layers, self-attention mechanisms, and point-wise feedforward networks. Thus, understanding the differences and interactions among these blocks is important. In this paper, we uncover a clear Sharpness Disparity across these blocks, which emerges early in training and intriguingly persists throughout the training process. Motivated by this finding, we propose Blockwise Learning Rate (LR), a strategy that tailors the LR to each block's sharpness, accelerating large language model (LLM) pre-training. By integrating Blockwise LR into AdamW, we consistently achieve lower terminal loss and nearly $2\times$ speedup compared to vanilla AdamW. We demonstrate this acceleration across GPT-2 and LLaMA, with model sizes ranging from 0.12B to 2B and datasets of OpenWebText, MiniPile, and C4. Finally, we incorporate Blockwise LR into Adam-mini (Zhang et al., 2024), a recently proposed memory-efficient variant of Adam, achieving a combined $2\times$ speedup and $2\times$ memory saving. These results underscore the potential of exploiting the sharpness disparity to improve LLM training.

Mingze Wang, Ruoxi Yu, Weinan E et al.

Transformers have demonstrated exceptional in-context learning capabilities, yet the theoretical understanding of the underlying mechanisms remains limited. A recent work (Elhage et al., 2021) identified a ``rich'' in-context mechanism known as induction head, contrasting with ``lazy'' $n$-gram models that overlook long-range dependencies. In this work, we provide both approximation and dynamics analyses of how transformers implement induction heads. In the {\em approximation} analysis, we formalize both standard and generalized induction head mechanisms, and examine how transformers can efficiently implement them, with an emphasis on the distinct role of each transformer submodule. For the {\em dynamics} analysis, we study the training dynamics on a synthetic mixed target, composed of a 4-gram and an in-context 2-gram component. This controlled setting allows us to precisely characterize the entire training process and uncover an {\em abrupt transition} from lazy (4-gram) to rich (induction head) mechanisms as training progresses.

Liangkai Hang, Junjie Yao, Zhiwei Bai et al.

The reasoning ability of large language models (LLMs) has been rapidly advancing in recent years, attracting interest in more fundamental approaches that can reliably enhance their generalizability. This work demonstrates that model complexity control, conveniently implementable by adjusting the initialization rate and weight decay coefficient, improves the scaling law of LLMs consistently over varying model sizes and data sizes. This gain is further illustrated by comparing the benchmark performance of 2.4B models pretrained on 1T tokens with different complexity hyperparameters. Instead of fixing the initialization std, we found that a constant initialization rate (the exponent of std) enables the scaling law to descend faster in both model and data sizes. These results indicate that complexity control is a promising direction for the continual advancement of LLMs.

Shuqi Lu, Haowei Lin, Lin Yao et al. · pku

3D structure modeling is essential across scales, enabling applications from fluid simulation and 3D reconstruction to protein folding and molecular docking. Yet, despite shared 3D spatial patterns, current approaches remain fragmented, with models narrowly specialized for specific domains and unable to generalize across tasks or scales. We propose Uni-3DAR, a unified autoregressive framework for cross-scale 3D generation and understanding. At its core is a coarse-to-fine tokenizer based on octree data structures, which compresses diverse 3D structures into compact 1D token sequences. We further propose a two-level subtree compression strategy, which reduces the octree token sequence by up to 8x. To address the challenge of dynamically varying token positions introduced by compression, we introduce a masked next-token prediction strategy that ensures accurate positional modeling, significantly boosting model performance. Extensive experiments across multiple 3D generation and understanding tasks, including small molecules, proteins, polymers, crystals, and macroscopic 3D objects, validate its effectiveness and versatility. Notably, Uni-3DAR surpasses previous state-of-the-art diffusion models by a substantial margin, achieving up to 256\% relative improvement while delivering inference speeds up to 21.8x faster.

Yingying Sun, Jun A, Zhiwei Liu et al.

Artificial intelligence (AI) is transforming scientific research, including proteomics. Advances in mass spectrometry (MS)-based proteomics data quality, diversity, and scale, combined with groundbreaking AI techniques, are unlocking new challenges and opportunities in biological discovery. Here, we highlight key areas where AI is driving innovation, from data analysis to new biological insights. These include developing an AI-friendly ecosystem for proteomics data generation, sharing, and analysis; improving peptide and protein identification and quantification; characterizing protein-protein interactions and protein complexes; advancing spatial and perturbation proteomics; integrating multi-omics data; and ultimately enabling AI-empowered virtual cells.

Yuzhi Liu, Duo Zhang, Anyang Peng et al.

Machine Learning Interatomic Potentials (MLIPs) enable accurate large-scale atomistic simulations, yet improving their expressive capacity efficiently remains challenging. Here we systematically develop Mixture-of-Experts (MoE) and Mixture-of-Linear-Experts (MoLE) architectures for MLIPs and analyze the effects of routing strategies and expert designs. We show that sparse activation combined with shared experts yields substantial performance gains, and that nonlinear MoE formulations outperform MoLE when shared experts are present, underscoring the importance of nonlinear expert specialization. Furthermore, element-wise routing consistently surpasses configuration-level routing, while global MoE routing often leads to numerical instability. The resulting element-wise MoE model achieves state-of-the-art accuracy across the OMol25, OMat24, and OC20M benchmarks. Analysis of routing patterns reveals chemically interpretable expert specialization aligned with periodic-table trends, indicating that the model effectively captures element-specific chemical characteristics for precise interatomic modeling.

Lirong Wu, Junjie Wang, Zhifeng Gao et al.

Organic reaction, the foundation of modern chemical industry, is crucial for new material development and drug discovery. However, deciphering reaction mechanisms and modeling multi-molecular relationships remain formidable challenges due to the complexity of molecular dynamics. While several state-of-the-art models like Uni-Mol2 have revolutionized single-molecular representation learning, their extension to multi-molecular systems, where chemical reactions inherently occur, has been underexplored. This paper introduces Uni-Mol3, a novel deep learning framework that employs a hierarchical pipeline for multi-molecular reaction modeling. At its core, Uni-Mol3 adopts a multi-scale molecular tokenizer (Mol-Tokenizer) that encodes 3D structures of molecules and other features into discrete tokens, creating a 3D-aware molecular language. The framework innovatively combines two pre-training stages: molecular pre-training to learn the molecular grammars and reaction pre-training to capture fundamental reaction principles, forming a progressive learning paradigm from single- to multi-molecular systems. With prompt-aware downstream fine-tuning, Uni-Mol3 demonstrates exceptional performance in diverse organic reaction tasks and supports multi-task prediction with strong generalizability. Experimental results across 10 datasets spanning 4 downstream tasks show that Uni-Mol3 outperforms existing methods, validating its effectiveness in modeling complex organic reactions. This work not only ushers in an alternative paradigm for multi-molecular computational modeling but also charts a course for intelligent organic reaction by bridging molecular representation with reaction mechanism understanding.

Haitao Lin, Junjie Wang, Zhifeng Gao et al.

The essence of a chemical reaction lies in the redistribution and reorganization of electrons, which is often manifested through electron transfer or the migration of electron pairs. These changes are inherently discrete and abrupt in the physical world, such as alterations in the charge states of atoms or the formation and breaking of chemical bonds. To model the transition of states, we propose SynBridge, a bidirectional flow-based generative model to achieve multi-task reaction prediction. By leveraging a graph-to-graph transformer network architecture and discrete flow bridges between any two discrete distributions, SynBridge captures bidirectional chemical transformations between graphs of reactants and products through the bonds' and atoms' discrete states. We further demonstrate the effectiveness of our method through extensive experiments on three benchmark datasets (USPTO-50K, USPTO-MIT, Pistachio), achieving state-of-the-art performance in both forward and retrosynthesis tasks. Our ablation studies and noise scheduling analysis reveal the benefits of structured diffusion over discrete spaces for reaction prediction.

Jiale Zhao, Pengzhi Mao, Kaifei Wang et al.

Deep learning has advanced mass spectrometry data interpretation, yet most models remain feature extractors rather than unified scoring frameworks. We present pUniFind, the first large-scale multimodal pre-trained model in proteomics that integrates end-to-end peptide-spectrum scoring with open, zero-shot de novo sequencing. Trained on over 100 million open search-derived spectra, pUniFind aligns spectral and peptide modalities via cross modality prediction and outperforms traditional engines across diverse datasets, particularly achieving a 42.6 percent increase in the number of identified peptides in immunopeptidomics. Supporting over 1,300 modifications, pUniFind identifies 60 percent more PSMs than existing de novo methods despite a 300-fold larger search space. A deep learning based quality control module further recovers 38.5 percent additional peptides including 1,891 mapped to the genome but absent from reference proteomes while preserving full fragment ion coverage. These results establish a unified, scalable deep learning framework for proteomic analysis, offering improved sensitivity, modification coverage, and interpretability.

Mingze Wang, Weinan E

Mixture-of-experts networks (MoEs) have demonstrated remarkable efficiency in modern deep learning. Despite their empirical success, the theoretical foundations underlying their ability to model complex tasks remain poorly understood. In this work, we conduct a systematic study of the expressive power of MoEs in modeling complex tasks with two common structural priors: low-dimensionality and sparsity. For shallow MoEs, we prove that they can efficiently approximate functions supported on low-dimensional manifolds, overcoming the curse of dimensionality. For deep MoEs, we show that $\cO(L)$-layer MoEs with $E$ experts per layer can approximate piecewise functions comprising $E^L$ pieces with compositional sparsity, i.e., they can exhibit an exponential number of structured tasks. Our analysis reveals the roles of critical architectural components and hyperparameters in MoEs, including the gating mechanism, expert networks, the number of experts, and the number of layers, and offers natural suggestions for MoE variants.

Mingze Wang, Jinbo Wang, Jiaqi Zhang et al.

We propose GradPower, a lightweight gradient-transformation technique for accelerating language model pre-training. Given a gradient vector $g=(g_i)_i$, GradPower first applies the elementwise sign-power transformation: $\varphi_p(g)=({\rm sign}(g_i)|g_i|^p)_{i}$ for a fixed $p>0$, and then feeds the transformed gradient into a base optimizer. Notably, GradPower requires only a single-line code change and no modifications to the base optimizer's internal logic, including the hyperparameters. When applied to Adam (termed AdamPower), GradPower consistently achieves lower terminal loss across diverse architectures (LLaMA, Qwen2MoE), parameter scales (66M to 2B), datasets (C4, OpenWebText), and learning-rate schedules (cosine, warmup-stable-decay). The most pronounced gains are observed when training modern mixture-of-experts models with warmup-stable-decay schedules. GradPower also integrates seamlessly with other state-of-the-art optimizers, such as Muon, yielding further improvements. Finally, we provide theoretical analyses that reveal the underlying mechanism of GradPower and highlights the influence of gradient noise.

Xiaohong Ji, Zhen Wang, Zhifeng Gao et al.

In recent years, pretraining models have made significant advancements in the fields of natural language processing (NLP), computer vision (CV), and life sciences. The significant advancements in NLP and CV are predominantly driven by the expansion of model parameters and data size, a phenomenon now recognized as the scaling laws. However, research exploring scaling law in molecular pretraining models remains unexplored. In this work, we present Uni-Mol2 , an innovative molecular pretraining model that leverages a two-track transformer to effectively integrate features at the atomic level, graph level, and geometry structure level. Along with this, we systematically investigate the scaling law within molecular pretraining models, characterizing the power-law correlations between validation loss and model size, dataset size, and computational resources. Consequently, we successfully scale Uni-Mol2 to 1.1 billion parameters through pretraining on 800 million conformations, making it the largest molecular pretraining model to date. Extensive experiments show consistent improvement in the downstream tasks as the model size grows. The Uni-Mol2 with 1.1B parameters also outperforms existing methods, achieving an average 27% improvement on the QM9 and 14% on COMPAS-1D dataset.

Zhongwang Zhang, Zhiwei Wang, Junjie Yao et al.

Understanding transformer-based language models is becoming increasingly crucial, particularly as they play pivotal roles in advancing towards artificial general intelligence. However, language model research faces significant challenges, especially for academic research groups with constrained resources. These challenges include complex data structures, unknown target functions, high computational costs and memory requirements, and a lack of interpretability in the inference process, etc. Drawing a parallel to the use of simple models in scientific research, we propose the concept of an anchor function. This is a type of benchmark function designed for studying language models in learning tasks that follow an "anchor-key" pattern. By utilizing the concept of an anchor function, we can construct a series of functions to simulate various language tasks. The anchor function plays a role analogous to that of mice in diabetes research, particularly suitable for academic research. We demonstrate the utility of the anchor function with an example, revealing two basic operations by attention structures in language models: shifting tokens and broadcasting one token from one position to many positions. These operations are also commonly observed in large language models. The anchor function framework, therefore, opens up a series of valuable and accessible research questions for further exploration, especially for theoretical study.

Jun Zhang, Xiaohan Lin, Weinan E et al.

Multiscale molecular modeling is widely applied in scientific research of molecular properties over large time and length scales. Two specific challenges are commonly present in multiscale modeling, provided that information between the coarse and fine representations of molecules needs to be properly exchanged: One is to construct coarse grained models by passing information from the fine to coarse levels; the other is to restore finer molecular details given coarse grained configurations. Although these two problems are commonly addressed independently, in this work, we present a theory connecting them, and develop a methodology called Cycle Coarse Graining (CCG) to solve both problems in a unified manner. In CCG, reconstruction can be achieved via a tractable deep generative model, allowing retrieval of fine details from coarse-grained simulations. The reconstruction in turn delivers better coarse-grained models which are informed of the fine-grained physics, and enables calculation of the free energies in a rare-event-free manner. CCG thus provides a systematic way for multiscale molecular modeling, where the finer details of coarse-grained simulations can be efficiently retrieved, and the coarse-grained models can be improved consistently.

Tianhan Zhang, Yuxiao Yi, Yifan Xu et al.

Machine learning has long been considered as a black box for predicting combustion chemical kinetics due to the extremely large number of parameters and the lack of evaluation standards and reproducibility. The current work aims to understand two basic questions regarding the deep neural network (DNN) method: what data the DNN needs and how general the DNN method can be. Sampling and preprocessing determine the DNN training dataset, further affect DNN prediction ability. The current work proposes using Box-Cox transformation (BCT) to preprocess the combustion data. In addition, this work compares different sampling methods with or without preprocessing, including the Monte Carlo method, manifold sampling, generative neural network method (cycle-GAN), and newly-proposed multi-scale sampling. Our results reveal that the DNN trained by the manifold data can capture the chemical kinetics in limited configurations but cannot remain robust toward perturbation, which is inevitable for the DNN coupled with the flow field. The Monte Carlo and cycle-GAN samplings can cover a wider phase space but fail to capture small-scale intermediate species, producing poor prediction results. A three-hidden-layer DNN, based on the multi-scale method without specific flame simulation data, allows predicting chemical kinetics in various scenarios and being stable during the temporal evolutions. This single DNN is readily implemented with several CFD codes and validated in various combustors, including (1). zero-dimensional autoignition, (2). one-dimensional freely propagating flame, (3). two-dimensional jet flame with triple-flame structure, and (4). three-dimensional turbulent lifted flames. The results demonstrate the satisfying accuracy and generalization ability of the pre-trained DNN. The Fortran and Python versions of DNN and example code are attached in the supplementary for reproducibility.

Zhiwei Wang, Yaoyu Zhang, Enhan Zhao et al.

A deep learning-based model reduction (DeePMR) method for simplifying chemical kinetics is proposed and validated using high-temperature auto-ignitions, perfectly stirred reactors (PSR), and one-dimensional freely propagating flames of n-heptane/air mixtures. The mechanism reduction is modeled as an optimization problem on Boolean space, where a Boolean vector, each entry corresponding to a species, represents a reduced mechanism. The optimization goal is to minimize the reduced mechanism size given the error tolerance of a group of pre-selected benchmark quantities. The key idea of the DeePMR is to employ a deep neural network (DNN) to formulate the objective function in the optimization problem. In order to explore high dimensional Boolean space efficiently, an iterative DNN-assisted data sampling and DNN training procedure are implemented. The results show that DNN-assistance improves sampling efficiency significantly, selecting only $10^5$ samples out of $10^{34}$ possible samples for DNN to achieve sufficient accuracy. The results demonstrate the capability of the DNN to recognize key species and reasonably predict reduced mechanism performance. The well-trained DNN guarantees the optimal reduced mechanism by solving an inverse optimization problem. By comparing ignition delay times, laminar flame speeds, temperatures in PSRs, the resulting skeletal mechanism has fewer species (45 species) but the same level of accuracy as the skeletal mechanism (56 species) obtained by the Path Flux Analysis (PFA) method. In addition, the skeletal mechanism can be further reduced to 28 species if only considering atmospheric, near-stoichiometric conditions (equivalence ratio between 0.6 and 1.2). The DeePMR provides an innovative way to perform model reduction and demonstrates the great potential of data-driven methods in the combustion area.

Lidong Fang, Pei Ge, Lei Zhang et al.

A long standing problem in the modeling of non-Newtonian hydrodynamics of polymeric flows is the availability of reliable and interpretable hydrodynamic models that faithfully encode the underlying micro-scale polymer dynamics. The main complication arises from the long polymer relaxation time, the complex molecular structure and heterogeneous interaction. DeePN$^2$, a deep learning-based non-Newtonian hydrodynamic model, has been proposed and has shown some success in systematically passing the micro-scale structural mechanics information to the macro-scale hydrodynamics for suspensions with simple polymer conformation and bond potential. The model retains a multi-scaled nature by mapping the polymer configurations into a set of symmetry-preserving macro-scale features. The extended constitutive laws for these macro-scale features can be directly learned from the kinetics of their micro-scale counterparts. In this paper, we develop DeePN$^2$ using more complex micro-structural models. We show that DeePN$^2$ can faithfully capture the broadly overlooked viscoelastic differences arising from the specific molecular structural mechanics without human intervention.

Jiequn Han, Yucheng Yang, Weinan E

An efficient, reliable, and interpretable global solution method, the Deep learning-based algorithm for Heterogeneous Agent Models (DeepHAM), is proposed for solving high dimensional heterogeneous agent models with aggregate shocks. The state distribution is approximately represented by a set of optimal generalized moments. Deep neural networks are used to approximate the value and policy functions, and the objective is optimized over directly simulated paths. In addition to being an accurate global solver, this method has three additional features. First, it is computationally efficient in solving complex heterogeneous agent models, and it does not suffer from the curse of dimensionality. Second, it provides a general and interpretable representation of the distribution over individual states, which is crucial in addressing the classical question of whether and how heterogeneity matters in macroeconomics. Third, it solves the constrained efficiency problem as easily as it solves the competitive equilibrium, which opens up new possibilities for studying optimal monetary and fiscal policies in heterogeneous agent models with aggregate shocks.

Lulu Zhang, Tao Luo, Yaoyu Zhang et al.

In this paper, we propose a a machine learning approach via model-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven by a model to solve PDEs based on operator representation with regularization from data. For linear PDEs, we use a DNN to parameterize the Green's function and obtain the neural operator to approximate the solution according to the Green's method. To train the DNN, the empirical risk consists of the mean squared loss with the least square formulation or the variational formulation of the governing equation and boundary conditions. For complicated problems, the empirical risk also includes a few labels, which are computed on coarse grid points with cheap computation cost and significantly improves the model accuracy. Intuitively, the labeled dataset works as a regularization in addition to the model constraints. The MOD-Net solves a family of PDEs rather than a specific one and is much more efficient than original neural operator because few expensive labels are required. We numerically show MOD-Net is very efficient in solving Poisson equation and one-dimensional radiative transfer equation. For nonlinear PDEs, the nonlinear MOD-Net can be similarly used as an ansatz for solving nonlinear PDEs, exemplified by solving several nonlinear PDE problems, such as the Burgers equation.

Hongkang Yang, Weinan E

The generative adversarial network (GAN) is a well-known model for learning high-dimensional distributions, but the mechanism for its generalization ability is not understood. In particular, GAN is vulnerable to the memorization phenomenon, the eventual convergence to the empirical distribution. We consider a simplified GAN model with the generator replaced by a density, and analyze how the discriminator contributes to generalization. We show that with early stopping, the generalization error measured by Wasserstein metric escapes from the curse of dimensionality, despite that in the long term, memorization is inevitable. In addition, we present a hardness of learning result for WGAN.

Jihao Long, Jiequn Han, Weinan E

Reinforcement learning (RL) algorithms based on high-dimensional function approximation have achieved tremendous empirical success in large-scale problems with an enormous number of states. However, most analysis of such algorithms gives rise to error bounds that involve either the number of states or the number of features. This paper considers the situation where the function approximation is made either using the kernel method or the two-layer neural network model, in the context of a fitted Q-iteration algorithm with explicit regularization. We establish an $\tilde{O}(H^3|\mathcal {A}|^{\frac14}n^{-\frac14})$ bound for the optimal policy with $Hn$ samples, where $H$ is the length of each episode and $|\mathcal {A}|$ is the size of action space. Our analysis hinges on analyzing the $L^2$ error of the approximated Q-function using $n$ data points. Even though this result still requires a finite-sized action space, the error bound is independent of the dimensionality of the state space.