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Theory · Everything Else

Francesco Mori, Stefano Sarao Mannelli, Francesca Mignacco

Artificial neural networks often struggle with _catastrophic forgetting_ when learning multiple tasks sequentially, as training on new tasks degrades the performance on previously learned tasks. Recent theoretical work has addressed this issue by analysing learning curves in synthetic frameworks under predefined training protocols. However, these protocols relied on heuristics and lacked a solid theoretical foundation assessing their optimality. In this paper, we fill this gap by combining exact equations for training dynamics, derived using statistical physics techniques, with optimal control methods. We apply this approach to teacher-student models for continual learning and multi-task problems, obtaining a theory for task-selection protocols maximising performance while minimising forgetting. Our theoretical analysis offers non-trivial yet interpretable strategies for mitigating catastrophic forgetting, shedding light on how optimal learning protocols modulate established effects, such as the influence of task similarity on forgetting. Finally, we validate our theoretical findings with experiments on real-world data.

Deep Learning · Everything Else

Leonardo Ferreira Guilhoto, Paris Perdikaris

This paper explores alternative formulations of the Kolmogorov Superposition Theorem (KST) as a foundation for neural network design. The original KST formulation, while mathematically elegant, presents practical challenges due to its limited insight into the structure of inner and outer functions and the large number of unknown variables it introduces. Kolmogorov-Arnold Networks (KANs) leverage KST for function approximation, but they have faced scrutiny due to mixed results compared to traditional multilayer perceptrons (MLPs) and practical limitations imposed by the original KST formulation. To address these issues, we introduce ActNet, a scalable deep learning model that builds on the KST and overcomes some of the drawbacks of Kolmogorov's original formulation. We evaluate ActNet in the context of Physics-Informed Neural Networks (PINNs), a framework well-suited for leveraging KST's strengths in low-dimensional function approximation, particularly for simulating partial differential equations (PDEs). In this challenging setting, where models must learn latent functions without direct measurements, ActNet consistently outperforms KANs across multiple benchmarks and is competitive against the current best MLP-based approaches. These results present ActNet as a promising new direction for KST-based deep learning applications, particularly in scientific computing and PDE simulation tasks.

Deep Learning · Algorithms

Abdolmehdi Behroozi, Chaopeng Shen, Daniel Kifer

Parametric differential equations of the form $\frac{\partial u}{\partial t} = f(u, x, t, p)$ are fundamental in science and engineering. While deep learning frameworks like the Fourier Neural Operator (FNO) efficiently approximate differential equation solutions, they struggle with inverse problems, sensitivity calculations $\frac{\partial u}{\partial p}$, and concept drift. We address these challenges by introducing a novel sensitivity loss regularizer, demonstrated through Sensitivity-Constrained Fourier Neural Operators (SC-FNO). Our approach maintains high accuracy for solution paths and outperforms both standard FNO and FNO with Physics-Informed Neural Network regularization. SC-FNO exhibits superior performance in parameter inversion tasks, accommodates more complex parameter spaces (tested with up to 82 parameters), reduces training data requirements, and decreases training time while maintaining accuracy. These improvements apply across various differential equations and neural operators, enhancing their reliability without significant computational overhead (30%–130% extra training time per epoch). Models and selected experiment code are available at: [https://github.com/AMBehroozi/SC_Neural_Operators](https://github.com/AMBehroozi/SC_Neural_Operators).

Applications · Computer Vision

Haiyang Jiang, Tetsuichi Wazawa, Imari Sato, Takeharu Nagai, Yinqiang Zheng

Electron-multiplying charge-coupled device (EMCCD) has been instrumental in sensitive observations under low-light situations including astronomy, material science, and biology. Despite its ingenious designs to enhance target signals overcoming read-out circuit noises, produced images are not completely noise free, which could still cast a cloud on desired experiment outcomes, especially in fluorescence microscopy.Existing studies on EMCCD's noise model have been focusing on statistical characteristics in theory, yet unable to incorporate latest advancements in the field of computational photography, where physics-based noise models are utilized to guide deep learning processes, creating adaptive denoising algorithms for ordinary image sensors.Still, those models are not directly applicable to EMCCD.In this paper, we intend to pioneer EMCCD denoising by introducing a systematic study on physics-based noise model calibration procedures for an EMCCD camera, accurately estimating statistical features of observable noise components in experiments, which are then utilized to generate substantial amount of authentic training samples for one of the most recent neural networks.A first real-world test image dataset for EMCCD is captured, containing both images of ordinary daily scenes and those of microscopic contents.Benchmarking upon the testset and authentic microscopic images, we demonstrate distinct advantages of our model against previous methods for EMCCD and physics-based noise modeling, forging a promising new path for EMCCD denoising.

Deep Learning · Generative Models and Autoencoders

Enrico Ventura, Beatrice Achilli, Gianluigi Silvestri, Carlo Lucibello, Luca Ambrogioni

In this paper, we investigate the latent geometry of generative diffusion models under the manifold hypothesis. For this purpose, we analyze the spectrum of eigenvalues (and singular values) of the Jacobian of the score function, whose discontinuities (gaps) reveal the presence and dimensionality of distinct sub-manifolds. Using a statistical physics approach, we derive the spectral distributions and formulas for the spectral gaps under several distributional assumptions, and we compare these theoretical predictions with the spectra estimated from trained networks. Our analysis reveals the existence of three distinct qualitative phases during the generative process: a trivial phase; a manifold coverage phase where the diffusion process fits the distribution internal to the manifold; a consolidation phase where the score becomes orthogonal to the manifold and all particles are projected on the support of the data. This `division of labor' between different timescales provides an elegant explanation of why generative diffusion models are not affected by the manifold overfitting phenomenon that plagues likelihood-based models, since the internal distribution and the manifold geometry are produced at different time points during generation.

Applications · Neuroscience, Cognitive Science

Will Dorrell, Kyle Hsu, Luke Hollingsworth, Jin Hwa Lee, Jiajun Wu, Chelsea Finn, Peter Latham, Timothy Behrens, James Whittington

Why do biological and artificial neurons sometimes modularise, each encoding a single meaningful variable, and sometimes entangle their representation of many variables? In this work, we develop a theory of when biologically inspired networks---those that are nonnegative and energy efficient---modularise their representation of source variables (sources). We derive necessary and sufficient conditions on a sample of sources that determine whether the neurons in an optimal biologically-inspired linear autoencoder modularise. Our theory applies to any dataset, extending far beyond the case of statistical independence studied in previous work. Rather we show that sources modularise if their support is ``sufficiently spread''. From this theory, we extract and validate predictions in a variety of empirical studies on how data distribution affects modularisation in nonlinear feedforward and recurrent neural networks trained on supervised and unsupervised tasks. Furthermore, we apply these ideas to neuroscience data, showing that range independence can be used to understand the mixing or modularising of spatial and reward information in entorhinal recordings in seemingly conflicting experiments. Further, we use these results to suggest alternate origins of mixed-selectivity, beyond the predominant theory of flexible nonlinear classification. In sum, our theory prescribes precise conditions on when neural activities modularise, providing tools for inducing and elucidating modular representations in brains and machines.

Deep Learning · Sequential Models, Time series

Filip Matzner, František Mráz

Echo State Networks (ESNs) are a class of recurrent neural networks in which only a small readout regression layer is trained, while the weights of the recurrent network, termed the reservoir, are randomly assigned and remain fixed. Our work introduces the Locally Connected ESN (LCESN), a novel ESN variant with a locally connected reservoir, forced memory, and a weight adaptation strategy. LCESN significantly reduces the asymptotic time and space complexities compared to the conventional ESN, enabling substantially larger networks. LCESN also improves the memory properties of ESNs without affecting network stability. We evaluate LCESN's performance on the NARMA10 benchmark task and compare it to state-of-the-art models on nine real-world datasets. Despite the simplicity of our model and its one-shot training approach, LCESN achieves competitive results, even surpassing several state-of-the-art models. LCESN introduces a fresh approach to real-world time series forecasting and demonstrates that large, well-tuned random recurrent networks can rival complex gradient-trained feedforward models. We provide our GPU-based implementation of LCESN as an open-source library.

Deep Learning · Other Representation Learning

Katharina Friedl, Noémie Jaquier, Jens Lundell, Tamim Asfour, Danica Kragic

By incorporating physical consistency as inductive bias, deep neural networks display increased generalization capabilities and data efficiency in learning nonlinear dynamic models. However, the complexity of these models generally increases with the system dimensionality, requiring larger datasets, more complex deep networks, and significant computational effort.We propose a novel geometric network architecture to learn physically-consistent reduced-order dynamic parameters that accurately describe the original high-dimensional system behavior.This is achieved by building on recent advances in model-order reduction and by adopting a Riemannian perspective to jointly learn a non-linear structure-preserving latent space and the associated low-dimensional dynamics.Our approach enables accurate long-term predictions of the high-dimensional dynamics of rigid and deformable systems with increased data efficiency by inferring interpretable and physically-plausible reduced Lagrangian models.

Applications · Genetics, Cell Biology, Health, etc

Lazar Atanackovic, Xi (Nicole) Zhang, Brandon Amos, Mathieu Blanchette, Leo J Lee, Yoshua Bengio, Alexander Tong, Kirill Neklyudov

Numerous biological and physical processes can be modeled as systems of interacting entities evolving continuously over time, e.g. the dynamics of communicating cells or physical particles. Learning the dynamics of such systems is essential for predicting the temporal evolution of populations across novel samples and unseen environments. Flow-based models allow for learning these dynamics at the population level - they model the evolution of the entire distribution of samples. However, current flow-based models are limited to a single initial population and a set of predefined conditions which describe different dynamics. We argue that multiple processes in natural sciences have to be represented as vector fields on the Wasserstein manifold of probability densities. That is, the change of the population at any moment in time depends on the population itself due to the interactions between samples. In particular, this is crucial for personalized medicine where the development of diseases and their respective treatment response depend on the microenvironment of cells specific to each patient. We propose *Meta Flow Matching* (MFM), a practical approach to integrate along these vector fields on the Wasserstein manifold by amortizing the flow model over the initial populations. Namely, we embed the population of samples using a Graph Neural Network (GNN) and use these embeddings to train a Flow Matching model. This gives MFM the ability to generalize over the initial distributions, unlike previously proposed methods. We demonstrate the ability of MFM to improve the prediction of individual treatment responses on a large-scale multi-patient single-cell drug screen dataset.

Applications · Chemistry and Drug Discovery

Jiarui Lu, Xiaoyin Chen, Stephen Lu, Chence Shi, Hongyu Guo, Yoshua Bengio, Jian Tang

Proteins adopt multiple structural conformations to perform their diverse biological functions, and understanding these conformations is crucial for advancing drug discovery. Traditional physics-based simulation methods often struggle with sampling equilibrium conformations and are computationally expensive. Recently, deep generative models have shown promise in generating protein conformations as a more efficient alternative. However, these methods predominantly rely on the diffusion process within a 3D geometric space, which typically centers around the vicinity of metastable states and is often inefficient in terms of runtime. In this paper, we introduce Structure Language Modeling (SLM) as a novel framework for efficient protein conformation generation. Specifically, the protein structures are first encoded into a compact latent space using a discrete variational auto-encoder, followed by conditional language modeling that effectively captures sequence-specific conformation distributions. This enables a more efficient and interpretable exploration of diverse ensemble modes compared to existing methods. Based on this general framework, we instantiate SLM with various popular LM architectures as well as proposing the ESMDiff, a novel BERT-like structure language model fine-tuned from ESM3 with masked diffusion. We verify our approach in various scenarios, including the equilibrium dynamics of BPTI, conformational change pairs, and intrinsically disordered proteins. SLM provides a highly efficient solution, offering a 20-100x speedup than existing methods in generating diverse conformations, shedding light on promising avenues for future research.

Applications · Computer Vision

Haiwen Feng, Michael J Black, Weiyang Liu, Peter Kulits, Victoria Abrevaya

Inverse graphics -- the task of inverting an image into physical variables that, when rendered, enable reproduction of the observed scene -- is a fundamental challenge in computer vision and graphics. Successfully disentangling an image into its constituent elements, such as the shape, color, and material properties of the objects of the 3D scene that produced it, requires a comprehensive understanding of the environment. This complexity limits the ability of existing carefully engineered approaches to generalize across domains. Inspired by the zero-shot ability of large language models (LLMs) to generalize to novel contexts, we investigate the possibility of leveraging the broad world knowledge encoded in such models to solve inverse-graphics problems. To this end, we propose the Inverse-Graphics Large Language Model (IG-LLM), an inverse-graphics framework centered around an LLM, that autoregressively decodes a visual embedding into a structured, compositional 3D-scene representation. We incorporate a frozen pre-trained visual encoder and a continuous numeric head to enable end-to-end training. Through our investigation, we demonstrate the potential of LLMs to facilitate inverse graphics through next-token prediction, without the application of image-space supervision. Our analysis enables new possibilities for precise spatial reasoning about images that exploit the visual knowledge of LLMs. We release our code and data at https://ig-llm.is.tue.mpg.de/ to ensure the reproducibility of our investigation and to facilitate future research.

Applications · Physics

Razmik Khosrovian, Takaharu Yaguchi, Hiroaki Yoshimura, Takashi Matsubara

Deep learning has achieved great success in modeling dynamical systems, providing data-driven simulators to predict complex phenomena, even without known governing equations. However, existing models have two major limitations: their narrow focus on mechanical systems and their tendency to treat systems as monolithic. These limitations reduce their applicability to dynamical systems in other domains, such as electrical and hydraulic systems, and to coupled systems. To address these limitations, we propose Poisson-Dirac Neural Networks (PoDiNNs), a novel framework based on the Dirac structure that unifies the port-Hamiltonian and Poisson formulations from geometric mechanics. This framework enables a unified representation of various dynamical systems across multiple domains as well as their interactions and degeneracies arising from couplings. Our experiments demonstrate that PoDiNNs offer improved accuracy and interpretability in modeling unknown coupled dynamical systems from data.

Deep Learning · Large Language Models

Clemencia Siro, Guy Gur-Ari, Gaurav Mishra, Stuart Shieber, Jason Phang, Zijie Wang, Kory Mathewson, Giorgio Mariani, Allen Nie, James Y Zou 等

Language models demonstrate both quantitative improvement and new qualitative capabilities with increasing scale. Despite their potentially transformative impact, these new capabilities are as yet poorly characterized. In order to inform future research, prepare for disruptive new model capabilities, and ameliorate socially harmful effects, it is vital that we understand the present and near-future capabilities and limitations of language models. To address this challenge, we introduce the Beyond the Imitation Game benchmark (BIG- bench). BIG-bench currently consists of 204 tasks, contributed by 450 authors across 132 institutions. Task topics are diverse, drawing problems from linguistics, childhood develop- ment, math, common-sense reasoning, biology, physics, social bias, software development, and beyond. BIG-bench focuses on tasks that are believed to be beyond the capabilities of current language models. We evaluate the behavior of OpenAI's GPT models, Google- internal dense transformer architectures, and Switch-style sparse transformers on BIG-bench, across model sizes spanning millions to hundreds of billions of parameters. In addition, a team of human expert raters performed all tasks in order to provide a strong baseline. Findings include: model performance and calibration both improve with scale, but are poor in absolute terms (and when compared with rater performance); performance is remarkably similar across model classes, though with benefits from sparsity; tasks that improve gradually and predictably commonly involve a large knowledge or memorization component, whereas tasks that exhibit "breakthrough" behavior at a critical scale often involve multiple steps or components, or brittle metrics; social bias typically increases with scale in settings with ambiguous context, but this can be improved with prompting.

Applications · Physics

Daniel Musekamp, Marimuthu Kalimuthu, David Holzmüller, Makoto Takamoto, Mathias Niepert

Solving partial differential equations (PDEs) is a fundamental problem in engineering and science. While neural PDE solvers can be more efficient than established numerical solvers, they often require large amounts of training data that is costly to obtain. Active learning (AL) could help surrogate models reach the same accuracy with smaller training sets by querying classical solvers with more informative initial conditions and PDE parameters. While AL is more common in other domains, it has yet to be studied extensively for neural PDE solvers. To bridge this gap, we introduce AL4PDE, a modular and extensible active learning benchmark. It provides multiple parametric PDEs and state-of-the-art surrogate models for the solver-in-the-loop setting, enabling the evaluation of existing and the development of new AL methods for PDE solving. We use the benchmark to evaluate batch active learning algorithms such as uncertainty- and feature-based methods. We show that AL reduces the average error by up to 71\% compared to random sampling and significantly reduces worst-case errors. Moreover, AL generates similar datasets across repeated runs, with consistent distributions over the PDE parameters and initial conditions. The acquired datasets are reusable, providing benefits for surrogate models not involved in the data generation.

Applications · Robotics

Yuejiang Liu, Jubayer Hamid, Annie Xie, Yoonho Lee, Max Du, Chelsea Finn

Predicting and executing a sequence of actions without intermediate replanning, known as action chunking, is increasingly used in robot learning from human demonstrations. Yet, its effects on the learned policy remain inconsistent: some studies find it crucial for achieving strong results, while others observe decreased performance. In this paper, we first dissect how action chunking impacts the divergence between a learner and a demonstrator. We find that action chunking allows the learner to better capture the temporal dependencies in demonstrations but at the cost of reduced reactivity to unexpected states. To address this tradeoff, we propose Bidirectional Decoding (BID), a test-time inference algorithm that bridges action chunking with closed-loop adaptation. At each timestep, BID samples multiple candidate predictions and searches for the optimal one based on two criteria: (i) backward coherence, which favors samples that align with previous decisions; (ii) forward contrast, which seeks samples of high likelihood for future plans. By coupling decisions within and across action chunks, BID promotes both long-term consistency and short-term reactivity. Experimental results show that our method boosts the performance of two state-of-the-art generative policies across seven simulation benchmarks and two real-world tasks. Code and videos are available at https://bid-robot.github.io.

Applications · Everything Else

Can Rong, Jingtao Ding, Yan Liu, Yong Li

Commuting Origin-Destination~(OD) flows are critical inputs for urban planning and transportation, providing crucial information about the population residing in one region and working in another within an interested area. Due to the high cost of data collection, researchers have developed physical and computational models to generate commuting OD flows using readily available urban attributes, such as sociodemographics and points of interest, for cities lacking historical OD flows \textemdash commuting OD flow generation. Existing works developed models based on different techniques and achieved improvement on different datasets with different evaluation metrics, which hinderes establishing a unified standard for comparing model performance. To bridge this gap, we introduce a large-scale dataset containing commuting OD flows for 3,333 areas including a wide range of urban environments around the United States. Based on that, we benchmark widely used models for commuting OD flow generation. We surprisingly find that the network-based generative models achieve the optimal performance in terms of both precision and generalization ability, which may inspire new research directions of graph generative modeling in this field. The dataset and benchmark are available at https://anonymous.4open.science/r/CommutingODGen-Dataset-0D4C/.

Applications · Physics

Tiexin Qin, Mengxu ZHU, Chunyang Li, Terry Lyons, Hong Yan, Haoliang Li

Understanding protein dynamics are essential for deciphering protein functional mechanisms and developing molecular therapies. However, the complex high-dimensional dynamics and interatomic interactions of biological processes pose significant challenge for existing computational techniques. In this paper, we approach this problem for the first time by introducing Deep Signature, a novel computationally tractable framework that characterizes complex dynamics and interatomic interactions based on their evolving trajectories. Specifically, our approach incorporates soft spectral clustering that locally aggregates cooperative dynamics to reduce the size of the system, as well as signature transform that collects iterated integrals to provide a global characterization of the non-smooth interactive dynamics. Theoretical analysis demonstrates that Deep Signature exhibits several desirable properties, including invariance to translation, near invariance to rotation, equivariance to permutation of atomic coordinates, and invariance under time reparameterization. Furthermore, experimental results on three benchmarks of biological processes verify that our approach can achieve superior performance compared to baseline methods.

Applications · Physics

Haoxiang Wang, Tao Yu, Hui Qiao, Qionghai Dai

Incompressible fluid on the surface is an interesting research area in the fluid simulation, which is the fundamental building block in visual effects, design of liquid crystal films, scientific analyses of atmospheric and oceanic phenomena, etc. The task brings two key challenges: the extension of the physical laws on 3D surfaces and the preservation of the energy and volume. Traditional methods rely on grids or meshes for spatial discretization, which leads to high memory consumption and a lack of robustness and adaptivity for various mesh qualities and representations. Many implicit representations based simulators like INSR are proposed for the storage efficiency and continuity, but they face challenges in the surface simulation and the energy dissipation. We propose a neural physical simulation framework on the surface with the implicit neural representation. Our method constructs a parameterized vector field with the exterior calculus and Closest Point Method on the surfaces, which guarantees the divergence-free property and enables the simulation on different surface representations (e.g. implicit neural represented surfaces). We further adopt a corresponding covariant derivative based advection process for surface flow dynamics and energy preservation. Our method shows higher accuracy, flexibility and memory-efficiency in the simulations of various surfaces with low energy dissipation. Numerical studies also highlight the potential of our framework across different practical applications such as vorticity shape generation and vector field Helmholtz decomposition.

Deep Learning · Large Language Models

Eunice Yiu, Maan Qraitem, Anisa Majhi, Charlie Wong, Yutong Bai, Shiry Ginosar, Alison Gopnik, Kate Saenko

This paper investigates visual analogical reasoning in large multimodal models (LMMs) compared to human adults and children. A “visual analogy” is an abstract rule inferred from one image and applied to another. While benchmarks exist for testing visual reasoning in LMMs, they require advanced skills and omit basic visual analogies that even young children can make. Inspired by developmental psychology, we propose a new benchmark of 4,300 visual transformations of everyday objects to test LMMs on visual analogical reasoning and compare them to children (ages three to five) and to adults. We structure the evaluation into three stages: identifying what changed (e.g., color, number, etc.), how it changed (e.g., added one object), and applying the rule to new scenarios. Our findings show that while GPT-o1, GPT-4V, LLaVA-1.5, and MANTIS identify the “what” effectively, they struggle with quantifying the “how” and extrapolating this rule to new objects. In contrast, children and adults exhibit much stronger analogical reasoning at all three stages. Additionally, the strongest tested model, GPT-o1, performs better in tasks involving simple surface-level visual attributes like color and size, correlating with quicker human adult response times. Conversely, more complex tasks such as number, rotation, and reflection, which necessitate extensive cognitive processing and understanding of extrinsic spatial properties in the physical world, present more significant challenges. Altogether, these findings highlight the limitations of training models on data that primarily consists of 2D images and text.

Applications · Physics

Youssef Shehata, Benjamin Holzschuh, Nils Thuerey

State-of-the-art Denoising Diffusion Probabilistic Models (DDPMs) rely on an expensive sampling process with a large Number of Function Evaluations (NFEs) to provide high-fidelity predictions. This computational bottleneck renders diffusion models less appealing as surrogates for the spatio-temporal prediction of physics-based problems with long rollout horizons. We propose Truncated Sampling Models, enabling single-step and few-step sampling with elevated fidelity by simple truncation of the diffusion process, reducing the gap between DDPMs and deterministic single-step approaches. We also introduce a novel approach, Iterative Refinement, to sample pre-trained DDPMs by reformulating the generative process as a refinement process with few sampling steps. Both proposed methods enable significant improvements in accuracy compared to DDPMs, DDIMs, and EDMs with NFEs $\leq$ 10 on a diverse set of experiments, including incompressible and compressible turbulent flow and airfoil flow uncertainty simulations. Our proposed methods provide stable predictions for long rollout horizons in time-dependent problems and are able to learn all modes of the data distribution in steady-state problems with high uncertainty.