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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
A recent numerical study observed that neural network classifiers enjoy a large degree of symmetry in the penultimate layer. Namely, if $h(x) = Af(x) +b$ where $A$ is a linear map and $f$ is the output of the penultimate layer of the network (after activation), then all data points $x_{i, 1}, \dots, x_{i, N_i}$ in a class $C_i$ are mapped to a single point $y_i$ by $f$ and the points $y_i$ are located at the vertices of a regular $k-1$-dimensional standard simplex in a high-dimensional Euclidean space. We explain this observation analytically in toy models for highly expressive deep neural networks. In complementary examples, we demonstrate rigorously that even the final output of the classifier $h$ is not uniform over data samples from a class $C_i$ if $h$ is a shallow network (or if the deeper layers do not bring the data samples into a convenient geometric configuration).
We use explicit representation formulas to show that solutions to certain partial differential equations lie in Barron spaces or multilayer spaces if the PDE data lie in such function spaces. Consequently, these solutions can be represented efficiently using artificial neural networks, even in high dimension. Conversely, we present examples in which the solution fails to lie in the function space associated to a neural network under consideration.
Models for learning probability distributions such as generative models and density estimators behave quite differently from models for learning functions. One example is found in the memorization phenomenon, namely the ultimate convergence to the empirical distribution, that occurs in generative adversarial networks (GANs). For this reason, the issue of generalization is more subtle than that for supervised learning. For the bias potential model, we show that dimension-independent generalization accuracy is achievable if early stopping is adopted, despite that in the long term, the model either memorizes the samples or diverges.
Developing efficient and accurate algorithms for chemistry integration is a challenging task due to its strong stiffness and high dimensionality. The current work presents a deep learning-based numerical method called DeepCombustion0.0 to solve stiff ordinary differential equation systems. The homogeneous autoignition of DME/air mixture, including 54 species, is adopted as an example to illustrate the validity and accuracy of the algorithm. The training and testing datasets cover a wide range of temperature, pressure, and mixture conditions between 750-1200 K, 30-50 atm, and equivalence ratio = 0.7-1.5. Both the first-stage low-temperature ignition (LTI) and the second-stage high-temperature ignition (HTI) are considered. The methodology highlights the importance of the adaptive data sampling techniques, power transform preprocessing, and binary deep neural network (DNN) design. By using the adaptive random samplings and appropriate power transforms, smooth submanifolds in the state vector phase space are observed, on which two three-layer DNNs can be appropriately trained. The neural networks are end-to-end, which predict temporal gradients of the state vectors directly. The results show that temporal evolutions predicted by DNN agree well with traditional numerical methods in all state vector dimensions, including temperature, pressure, and species concentrations. Besides, the ignition delay time differences are within 1%. At the same time, the CPU time is reduced by more than 20 times and 200 times compared with the HMTS and VODE method, respectively. The current work demonstrates the enormous potential of applying the deep learning algorithm in chemical kinetics and combustion modeling.
It is not clear yet why ADAM-alike adaptive gradient algorithms suffer from worse generalization performance than SGD despite their faster training speed. This work aims to provide understandings on this generalization gap by analyzing their local convergence behaviors. Specifically, we observe the heavy tails of gradient noise in these algorithms. This motivates us to analyze these algorithms through their Levy-driven stochastic differential equations (SDEs) because of the similar convergence behaviors of an algorithm and its SDE. Then we establish the escaping time of these SDEs from a local basin. The result shows that (1) the escaping time of both SGD and ADAM~depends on the Radon measure of the basin positively and the heaviness of gradient noise negatively; (2) for the same basin, SGD enjoys smaller escaping time than ADAM, mainly because (a) the geometry adaptation in ADAM~via adaptively scaling each gradient coordinate well diminishes the anisotropic structure in gradient noise and results in larger Radon measure of a basin; (b) the exponential gradient average in ADAM~smooths its gradient and leads to lighter gradient noise tails than SGD. So SGD is more locally unstable than ADAM~at sharp minima defined as the minima whose local basins have small Radon measure, and can better escape from them to flatter ones with larger Radon measure. As flat minima here which often refer to the minima at flat or asymmetric basins/valleys often generalize better than sharp ones , our result explains the better generalization performance of SGD over ADAM. Finally, experimental results confirm our heavy-tailed gradient noise assumption and theoretical affirmation.
The lack of interpretability and transparency are preventing economists from using advanced tools like neural networks in their empirical research. In this paper, we propose a class of interpretable neural network models that can achieve both high prediction accuracy and interpretability. The model can be written as a simple function of a regularized number of interpretable features, which are outcomes of interpretable functions encoded in the neural network. Researchers can design different forms of interpretable functions based on the nature of their tasks. In particular, we encode a class of interpretable functions named persistent change filters in the neural network to study time series cross-sectional data. We apply the model to predicting individual's monthly employment status using high-dimensional administrative data. We achieve an accuracy of 94.5% in the test set, which is comparable to the best performed conventional machine learning methods. Furthermore, the interpretability of the model allows us to understand the mechanism that underlies the prediction: an individual's employment status is closely related to whether she pays different types of insurances. Our work is a useful step towards overcoming the black-box problem of neural networks, and provide a new tool for economists to study administrative and proprietary big data.
The current knowledge system of macroeconomics is built on interactions among a small number of variables, since traditional macroeconomic models can mostly handle a handful of inputs. Recent work using big data suggests that a much larger number of variables are active in driving the dynamics of the aggregate economy. In this paper, we introduce a knowledge graph (KG) that consists of not only linkages between traditional economic variables but also new alternative big data variables. We extract these new variables and the linkages by applying advanced natural language processing (NLP) tools on the massive textual data of academic literature and research reports. As one example of the potential applications, we use it as the prior knowledge to select variables for economic forecasting models in macroeconomics. Compared to statistical variable selection methods, KG-based methods achieve significantly higher forecasting accuracy, especially for long run forecasts.
We consider binary and multi-class classification problems using hypothesis classes of neural networks. For a given hypothesis class, we use Rademacher complexity estimates and direct approximation theorems to obtain a priori error estimates for regularized loss functionals.
The purpose of this article is to review the achievements made in the last few years towards the understanding of the reasons behind the success and subtleties of neural network-based machine learning. In the tradition of good old applied mathematics, we will not only give attention to rigorous mathematical results, but also the insight we have gained from careful numerical experiments as well as the analysis of simplified models. Along the way, we also list the open problems which we believe to be the most important topics for further study. This is not a complete overview over this quickly moving field, but we hope to provide a perspective which may be helpful especially to new researchers in the area.
We study the approximation properties and optimization dynamics of recurrent neural networks (RNNs) when applied to learn input-output relationships in temporal data. We consider the simple but representative setting of using continuous-time linear RNNs to learn from data generated by linear relationships. Mathematically, the latter can be understood as a sequence of linear functionals. We prove a universal approximation theorem of such linear functionals, and characterize the approximation rate and its relation with memory. Moreover, we perform a fine-grained dynamical analysis of training linear RNNs, which further reveal the intricate interactions between memory and learning. A unifying theme uncovered is the non-trivial effect of memory, a notion that can be made precise in our framework, on approximation and optimization: when there is long term memory in the target, it takes a large number of neurons to approximate it. Moreover, the training process will suffer from slow downs. In particular, both of these effects become exponentially more pronounced with memory - a phenomenon we call the "curse of memory". These analyses represent a basic step towards a concrete mathematical understanding of new phenomenon that may arise in learning temporal relationships using recurrent architectures.
The dynamic behavior of RMSprop and Adam algorithms is studied through a combination of careful numerical experiments and theoretical explanations. Three types of qualitative features are observed in the training loss curve: fast initial convergence, oscillations, and large spikes in the late phase. The sign gradient descent (signGD) flow, which is the limit of Adam when taking the learning rate to 0 while keeping the momentum parameters fixed, is used to explain the fast initial convergence. For the late phase of Adam, three different types of qualitative patterns are observed depending on the choice of the hyper-parameters: oscillations, spikes, and divergence. In particular, Adam converges much smoother and even faster when the values of the two momentum factors are close to each other. This observation is particularly important for scientific computing tasks, for which the training process usually proceeds into the high precision regime.
We propose a systematic method for learning stable and physically interpretable dynamical models using sampled trajectory data from physical processes based on a generalized Onsager principle. The learned dynamics are autonomous ordinary differential equations parameterized by neural networks that retain clear physical structure information, such as free energy, diffusion, conservative motion and external forces. For high dimensional problems with a low dimensional slow manifold, an autoencoder with metric preserving regularization is introduced to find the low dimensional generalized coordinates on which we learn the generalized Onsager dynamics. Our method exhibits clear advantages over existing methods on benchmark problems for learning ordinary differential equations. We further apply this method to study Rayleigh-Benard convection and learn Lorenz-like low dimensional autonomous reduced order models that capture both qualitative and quantitative properties of the underlying dynamics. This forms a general approach to building reduced order models for forced dissipative systems.
The random feature model exhibits a kind of resonance behavior when the number of parameters is close to the training sample size. This behavior is characterized by the appearance of large generalization gap, and is due to the occurrence of very small eigenvalues for the associated Gram matrix. In this paper, we examine the dynamic behavior of the gradient descent algorithm in this regime. We show, both theoretically and experimentally, that there is a dynamic self-correction mechanism at work: The larger the eventual generalization gap, the slower it develops, both because of the small eigenvalues. This gives us ample time to stop the training process and obtain solutions with good generalization property.
We propose a general machine learning-based framework for building an accurate and widely-applicable energy functional within the framework of generalized Kohn-Sham density functional theory. To this end, we develop a way of training self-consistent models that are capable of taking large datasets from different systems and different kinds of labels. We demonstrate that the functional that results from this training procedure gives chemically accurate predictions on energy, force, dipole, and electron density for a large class of molecules. It can be continuously improved when more and more data are available.
We develop Banach spaces for ReLU neural networks of finite depth $L$ and infinite width. The spaces contain all finite fully connected $L$-layer networks and their $L^2$-limiting objects under bounds on the natural path-norm. Under this norm, the unit ball in the space for $L$-layer networks has low Rademacher complexity and thus favorable generalization properties. Functions in these spaces can be approximated by multi-layer neural networks with dimension-independent convergence rates. The key to this work is a new way of representing functions in some form of expectations, motivated by multi-layer neural networks. This representation allows us to define a new class of continuous models for machine learning. We show that the gradient flow defined this way is the natural continuous analog of the gradient descent dynamics for the associated multi-layer neural networks. We show that the path-norm increases at most polynomially under this continuous gradient flow dynamics.
We propose the coarse-grained spectral projection method (CGSP), a deep learning-assisted approach for tackling quantum unitary dynamic problems with an emphasis on quench dynamics. We show CGSP can extract spectral components of many-body quantum states systematically with sophisticated neural network quantum ansatz. CGSP exploits fully the linear unitary nature of the quantum dynamics, and is potentially superior to other quantum Monte Carlo methods for ergodic dynamics. Preliminary numerical results on 1D XXZ models with periodic boundary condition are carried out to demonstrate the practicality of CGSP.
A numerical and phenomenological study of the gradient descent (GD) algorithm for training two-layer neural network models is carried out for different parameter regimes when the target function can be accurately approximated by a relatively small number of neurons. It is found that for Xavier-like initialization, there are two distinctive phases in the dynamic behavior of GD in the under-parametrized regime: An early phase in which the GD dynamics follows closely that of the corresponding random feature model and the neurons are effectively quenched, followed by a late phase in which the neurons are divided into two groups: a group of a few "activated" neurons that dominate the dynamics and a group of background (or "quenched") neurons that support the continued activation and deactivation process. This neural network-like behavior is continued into the mildly over-parametrized regime, where it undergoes a transition to a random feature-like behavior. The quenching-activation process seems to provide a clear mechanism for "implicit regularization". This is qualitatively different from the dynamics associated with the "mean-field" scaling where all neurons participate equally and there does not appear to be qualitative changes when the network parameters are changed.
We study the natural function space for infinitely wide two-layer neural networks with ReLU activation (Barron space) and establish different representation formulae. In two cases, we describe the space explicitly up to isomorphism. Using a convenient representation, we study the pointwise properties of two-layer networks and show that functions whose singular set is fractal or curved (for example distance functions from smooth submanifolds) cannot be represented by infinitely wide two-layer networks with finite path-norm. We use this structure theorem to show that the only $C^1$-diffeomorphisms which Barron space are affine. Furthermore, we show that every Barron function can be decomposed as the sum of a bounded and a positively one-homogeneous function and that there exist Barron functions which decay rapidly at infinity and are globally Lebesgue-integrable. This result suggests that two-layer neural networks may be able to approximate a greater variety of functions than commonly believed.
Machine learning is poised as a very powerful tool that can drastically improve our ability to carry out scientific research. However, many issues need to be addressed before this becomes a reality. This article focuses on one particular issue of broad interest: How can we integrate machine learning with physics-based modeling to develop new interpretable and truly reliable physical models? After introducing the general guidelines, we discuss the two most important issues for developing machine learning-based physical models: Imposing physical constraints and obtaining optimal datasets. We also provide a simple and intuitive explanation for the fundamental reasons behind the success of modern machine learning, as well as an introduction to the concurrent machine learning framework needed for integrating machine learning with physics-based modeling. Molecular dynamics and moment closure of kinetic equations are used as examples to illustrate the main issues discussed. We end with a general discussion on where this integration will lead us to, and where the new frontier will be after machine learning is successfully integrated into scientific modeling.
We prove that the gradient descent training of a two-layer neural network on empirical or population risk may not decrease population risk at an order faster than $t^{-4/(d-2)}$ under mean field scaling. Thus gradient descent training for fitting reasonably smooth, but truly high-dimensional data may be subject to the curse of dimensionality. We present numerical evidence that gradient descent training with general Lipschitz target functions becomes slower and slower as the dimension increases, but converges at approximately the same rate in all dimensions when the target function lies in the natural function space for two-layer ReLU networks.
We establish a scale separation of Kolmogorov width type between subspaces of a given Banach space under the condition that a sequence of linear maps converges much faster on one of the subspaces. The general technique is then applied to show that reproducing kernel Hilbert spaces are poor $L^2$-approximators for the class of two-layer neural networks in high dimension, and that multi-layer networks with small path norm are poor approximators for certain Lipschitz functions, also in the $L^2$-topology.
We introduce a machine-learning-based framework for constructing continuum non-Newtonian fluid dynamics model directly from a micro-scale description. Dumbbell polymer solutions are used as examples to demonstrate the essential ideas. To faithfully retain molecular fidelity, we establish a micro-macro correspondence via a set of encoders for the micro-scale polymer configurations and their macro-scale counterparts, a set of nonlinear conformation tensors. The dynamics of these conformation tensors can be derived from the micro-scale model and the relevant terms can be parametrized using machine learning. The final model named the deep non-Newtonian model (DeePN$^2$), takes the form of conventional non-Newtonian fluid dynamics models, with a new form of the objective tensor derivative. Both the formulation of the dynamic equation and the neural network representation rigorously preserve the rotational invariance, which ensures the admissibility of the constructed model. Numerical results demonstrate the accuracy of DeePN$^2$, where models based on empirical closures show limitations.
We study the generalization properties of minimum-norm solutions for three over-parametrized machine learning models including the random feature model, the two-layer neural network model and the residual network model. We proved that for all three models, the generalization error for the minimum-norm solution is comparable to the Monte Carlo rate, up to some logarithmic terms, as long as the models are sufficiently over-parametrized.
One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation errors by a particular machine learning model. In this paper, we address this issue for two representative neural network models: the two-layer networks and the residual neural networks. We define the Barron space and show that it is the right space for two-layer neural network models in the sense that optimal direct and inverse approximation theorems hold for functions in the Barron space. For residual neural network models, we construct the so-called flow-induced function space, and prove direct and inverse approximation theorems for this space. In addition, we show that the Rademacher complexity for bounded sets under these norms has the optimal upper bounds.
The behavior of the gradient descent (GD) algorithm is analyzed for a deep neural network model with skip-connections. It is proved that in the over-parametrized regime, for a suitable initialization, with high probability GD can find a global minimum exponentially fast. Generalization error estimates along the GD path are also established. As a consequence, it is shown that when the target function is in the reproducing kernel Hilbert space (RKHS) with a kernel defined by the initialization, there exist generalizable early-stopping solutions along the GD path. In addition, it is also shown that the GD path is uniformly close to the functions given by the related random feature model. Consequently, in this "implicit regularization" setting, the deep neural network model deteriorates to a random feature model. Our results hold for neural networks of any width larger than the input dimension.
A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are updated. General initialization schemes as well as general regimes for the network width and training data size are considered. In the over-parametrized regime, it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the labels. In addition, it is proved that throughout the training process the functions represented by the neural network model are uniformly close to that of a kernel method. For general values of the network width and training data size, sharp estimates of the generalization error is established for target functions in the appropriate reproducing kernel Hilbert space.
Optimal a priori estimates are derived for the population risk, also known as the generalization error, of a regularized residual network model. An important part of the regularized model is the usage of a new path norm, called the weighted path norm, as the regularization term. The weighted path norm treats the skip connections and the nonlinearities differently so that paths with more nonlinearities are regularized by larger weights. The error estimates are a priori in the sense that the estimates depend only on the target function, not on the parameters obtained in the training process. The estimates are optimal, in a high dimensional setting, in the sense that both the bound for the approximation and estimation errors are comparable to the Monte Carlo error rates. A crucial step in the proof is to establish an optimal bound for the Rademacher complexity of the residual networks. Comparisons are made with existing norm-based generalization error bounds.
We develop the mathematical foundations of the stochastic modified equations (SME) framework for analyzing the dynamics of stochastic gradient algorithms, where the latter is approximated by a class of stochastic differential equations with small noise parameters. We prove that this approximation can be understood mathematically as an weak approximation, which leads to a number of precise and useful results on the approximations of stochastic gradient descent (SGD), momentum SGD and stochastic Nesterov's accelerated gradient method in the general setting of stochastic objectives. We also demonstrate through explicit calculations that this continuous-time approach can uncover important analytical insights into the stochastic gradient algorithms under consideration that may not be easy to obtain in a purely discrete-time setting.
New estimates for the population risk are established for two-layer neural networks. These estimates are nearly optimal in the sense that the error rates scale in the same way as the Monte Carlo error rates. They are equally effective in the over-parametrized regime when the network size is much larger than the size of the dataset. These new estimates are a priori in nature in the sense that the bounds depend only on some norms of the underlying functions to be fitted, not the parameters in the model, in contrast with most existing results which are a posteriori in nature. Using these a priori estimates, we provide a perspective for understanding why two-layer neural networks perform better than the related kernel methods.
We present a deep generative model, named Monge-Ampère flow, which builds on continuous-time gradient flow arising from the Monge-Ampère equation in optimal transport theory. The generative map from the latent space to the data space follows a dynamical system, where a learnable potential function guides a compressible fluid to flow towards the target density distribution. Training of the model amounts to solving an optimal control problem. The Monge-Ampère flow has tractable likelihoods and supports efficient sampling and inference. One can easily impose symmetry constraints in the generative model by designing suitable scalar potential functions. We apply the approach to unsupervised density estimation of the MNIST dataset and variational calculation of the two-dimensional Ising model at the critical point. This approach brings insights and techniques from Monge-Ampère equation, optimal transport, and fluid dynamics into reversible flow-based generative models.