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Deep Learning · Theory

Jeongjin Park, Nicole Yang, Nisha Chandramoorthy

Conventional notions of generalization often fail to describe the ability of learned models to capture meaningful information from dynamical data. A neural network that learns complex dynamics with a small test error may still fail to reproduce its \emph{physical} behavior, including associated statistical moments and Lyapunov exponents. To address this gap, we propose an ergodic theoretic approach to generalization of complex dynamical models learned from time series data. Our main contribution is to define and analyze generalization of a broad suite of neural representations of classes of ergodic systems, including chaotic systems, in a way that captures emulating underlying invariant, physical measures. Our results provide theoretical justification for why regression methods for generators of dynamical systems (Neural ODEs) fail to generalize, and why their statistical accuracy improves upon adding Jacobian information during training. We verify our results on a number of ergodic chaotic systems and neural network parameterizations, including MLPs, ResNets, Fourier Neural layers, and RNNs.

Physical Models · Physics

Philipp Schleich, Marta Skreta, Lasse Kristensen, Rodrigo Vargas-Hernandez, Alan Aspuru-Guzik

The feasibility of variational quantum algorithms, the most popular correspondent of neural networks on noisy, near-term quantum hardware, is highly impacted by the circuit depth of the involved parametrized quantum circuits (PQCs). Higher depth increases expressivity, but also results in a detrimental accumulation of errors. Furthermore, the number of parameters involved in the PQC significantly influences the performance through the necessary number of measurements to evaluate gradients, which scales linearly with the number of parameters. Motivated by this, we look at deep equilibrium models (DEQs), which mimic an infinite-depth, weight-tied network using a fraction of the memory by employing a root solver to find the fixed points of the network. In this work, we present Quantum Deep Equilibrium Models (QDEQs): a training paradigm that learns parameters of a quantum machine learning model given by a PQC using DEQs. To our knowledge, no work has yet explored the application of DEQs to QML models. We apply QDEQs to find the parameters of a quantum circuit in two settings: the first involves classifying MNIST-4 digits with 4 qubits; the second extends it to 10 classes of MNIST, FashionMNIST and CIFAR. We find that QDEQ is not only competitive with comparable existing baseline models, but also achieves higher performance than a network with 5 times more layers. This demonstrates that the QDEQ paradigm can be used to develop significantly more shallow quantum circuits for a given task, something which is essential for the utility of near-term quantum computers. Our code is available at \url{https://github.com/martaskrt/qdeq}.

Statistical Methods

Hamidreza Hashempoorikderi, Wan Choi

This paper advances temporal reasoning within dynamically changing high-dimensional noisy observations, focusing on a latent space that characterizes the nonlinear dynamics of objects in their environment. We introduce the *Gated Inference Network* (GIN), an efficient approximate Bayesian inference algorithm for state space models (SSMs) with nonlinear state transitions and emissions. GIN disentangles two latent representations: one representing the object derived from a nonlinear mapping model, and another representing the latent state describing its dynamics. This disentanglement enables direct state estimation and missing data imputation as the world evolves. To infer the latent state, we utilize a deep extended Kalman filter (EKF) approach that integrates a novel compact RNN structure to compute both the Kalman Gain (KG) and smoothing gain (SG), completing the data flow. This design results in a computational cost per step that is linearly faster than EKF but introduces issues such as the exploding gradient problem. To mitigate the exploding gradients caused by the compact RNN structure in our model, we propose a specialized learning method that ensures stable training and inference. The model is then trained end-to-end on videos depicting a diverse range of simulated and real-world physical systems, and outperforms its ounterparts —RNNs, autoregressive models, and variational approaches— in state estimation and missing data imputation tasks.

Applications

Keying Kuang, Frances Dean, Jack B. Jedlicki, David Ouyang, Anthony Philippakis, David Sontag, Ahmed Alaa

A digital twin is a virtual replica of a real-world physical phenomena that uses mathematical modeling to characterize and simulate its defining features. By constructing digital twins for disease processes, we can perform in-silico simulations that mimic patients' health conditions and counterfactual outcomes under hypothetical interventions in a virtual setting. This eliminates the need for invasive procedures or uncertain treatment decisions. In this paper, we propose a method to identify digital twin model parameters using only noninvasive patient health data. We approach the digital twin modeling as a composite inverse problem, and observe that its structure resembles pretraining and finetuning in self-supervised learning (SSL). Leveraging this, we introduce a physics-informed SSL algorithm that initially pretrains a neural network on the pretext task of learning a differentiable simulator of a physiological process. Subsequently, the model is trained to reconstruct physiological measurements from noninvasive modalities while being constrained by the physical equations learned in pretraining. We apply our method to identify digital twins of cardiac hemodynamics using noninvasive echocardiogram videos, and demonstrate its utility in unsupervised disease detection and in-silico clinical trials.

Haixu Wu, Huakun Luo, Yuezhou Ma, Jianmin Wang, Mingsheng Long

Physics-informed neural networks (PINNs) have been widely applied to solve partial differential equations (PDEs) by enforcing outputs and gradients of deep models to satisfy target equations. Due to the limitation of numerical computation, PINNs are conventionally optimized on finite selected points. However, since PDEs are usually defined on continuous domains, solely optimizing models on scattered points may be insufficient to obtain an accurate solution for the whole domain. To mitigate this inherent deficiency of the default scatter-point optimization, this paper proposes and theoretically studies a new training paradigm as region optimization. Concretely, we propose to extend the optimization process of PINNs from isolated points to their continuous neighborhood regions, which can theoretically decrease the generalization error, especially for hidden high-order constraints of PDEs. A practical training algorithm, Region Optimized PINN (RoPINN), is seamlessly derived from this new paradigm, which is implemented by a straightforward but effective Monte Carlo sampling method. By calibrating the sampling process into trust regions, RoPINN finely balances optimization and generalization error. Experimentally, RoPINN consistently boosts the performance of diverse PINNs on a wide range of PDEs without extra backpropagation or gradient calculation. Code is available at this repository: https://github.com/thuml/RoPINN.

Physical Models · Geoscience

Joel Oskarsson, Tomas Landelius, Marc Deisenroth, Fredrik Lindsten

In recent years, machine learning has established itself as a powerful tool for high-resolution weather forecasting. While most current machine learning models focus on deterministic forecasts, accurately capturing the uncertainty in the chaotic weather system calls for probabilistic modeling. We propose a probabilistic weather forecasting model called Graph-EFM, combining a flexible latent-variable formulation with the successful graph-based forecasting framework. The use of a hierarchical graph construction allows for efficient sampling of spatially coherent forecasts. Requiring only a single forward pass per time step, Graph-EFM allows for fast generation of arbitrarily large ensembles. We experiment with the model on both global and limited area forecasting. Ensemble forecasts from Graph-EFM achieve equivalent or lower errors than comparable deterministic models, with the added benefit of accurately capturing forecast uncertainty.

Physical Models

Hanqing Zhu, Wenyan Cong, Guojin Chen, Shupeng Ning, Ray Chen, Jiaqi Gu, David Z. Pan

Electromagnetic field simulation is central to designing, optimizing, and validating photonic devices and circuits. However, costly computation associated with numerical simulation poses a significant bottleneck, hindering scalability and turnaround time in the photonic circuit design process.Neural operators offer a promising alternative, but existing SOTA approaches, Neurolight, struggle with predicting high-fidelity fields for real-world complicated photonic devices, with the best reported 0.38 normalized mean absolute error in Neurolight.The interplays of highly complex light-matter interaction, e.g., scattering and resonance, sensitivity to local structure details, non-uniform learning complexity for full-domain simulation, and rich frequency information, contribute to the failure of existing neural PDE solvers.In this work, we boost the prediction fidelity to an unprecedented level for simulating complex photonic devices with a novel operator design driven by the above challenges.We propose a novel cross-axis factorized PACE operator with a strong long-distance modeling capacity to connect the full-domain complex field pattern with local device structures.Inspired by human learning, we further divide and conquer the simulation task for extremely hard cases into two progressively easy tasks, with a first-stage model learning an initial solution refined by a second model.On various complicated photonic device benchmarks, we demonstrate one sole PACE model is capable of achieving 73% lower error with 50% fewer parameters compared with various recent ML for PDE solvers.The two-stage setup further advances high-fidelity simulation for even more intricate cases.In terms of runtime, PACE demonstrates 154-577x and 11.8-12x simulation speedup over numerical solver using scipy or highly-optimized pardiso solver, respectively.We open-sourced the code and *complicated* optical device dataset at [PACE-Light](https://github.com/zhuhanqing/PACE-Light).

Tycho van der Ouderaa, Mark van der Wilk, Pim de Haan

Symmetries have proven useful in machine learning models, improving generalisation and overall performance. At the same time, recent advancements in learning dynamical systems rely on modelling the underlying Hamiltonian to guarantee the conservation of energy.These approaches can be connected via a seminal result in mathematical physics: Noether's theorem, which states that symmetries in a dynamical system correspond to conserved quantities.This work uses Noether's theorem to parameterise symmetries as learnable conserved quantities. We then allow conserved quantities and associated symmetries to be learned directly from train data through approximate Bayesian model selection, jointly with the regular training procedure. As training objective, we derive a variational lower bound to the marginal likelihood. The objective automatically embodies an Occam's Razor effect that avoids collapse of conversation laws to the trivial constant, without the need to manually add and tune additional regularisers. We demonstrate a proof-of-principle on n-harmonic oscillators and n-body systems. We find that our method correctly identifies the correct conserved quantities and U(n) and SE(n) symmetry groups, improving overall performance and predictive accuracy on test data.

Jiaxian Yan, ZAIXI ZHANG, Jintao Zhu, Kai Zhang, Jianfeng Pei, Qi Liu

Molecular docking, a technique for predicting ligand binding poses, is crucial in structure-based drug design for understanding protein-ligand interactions. Recent advancements in docking methods, particularly those leveraging geometric deep learning (GDL), have demonstrated significant efficiency and accuracy advantages over traditional sampling methods. Despite these advancements, current methods are often tailored for specific docking settings, and limitations such as the neglect of protein side-chain structures, difficulties in handling large binding pockets, and challenges in predicting physically valid structures exist. To accommodate various docking settings and achieve accurate, efficient, and physically reliable docking, we propose a novel two-stage docking framework, DeltaDock, consisting of pocket prediction and site-specific docking. We innovatively reframe the pocket prediction task as a pocket-ligand alignment problem rather than direct prediction in the first stage. Then we follow a bi-level coarse-to-fine iterative refinement process to perform site-specific docking. Comprehensive experiments demonstrate the superior performance of DeltaDock. Notably, in the blind docking setting, DeltaDock achieves a 31\% relative improvement over the docking success rate compared with the previous state-of-the-art GDL model DiffDock. With the consideration of physical validity, this improvement increases to about 300\%.

Physical Models

Mengyi Chen, Qianxiao Li

Macroscopic observables of a system are of keen interest in real applications such as the design of novel materials. Current methods rely on microscopic trajectory simulations, where the forces on all microscopic coordinates need to be computed or measured. However, this can be computationally prohibitive for realistic systems. In this paper, we propose a method to learn macroscopic dynamics requiring only force computations on a subset of the microscopic coordinates. Our method relies on a sparsity assumption: the force on each microscopic coordinate relies only on a small number of other coordinates. The main idea of our approach is to map the training procedure on the macroscopic coordinates back to the microscopic coordinates, on which partial force computations can be used as stochastic estimation to update model parameters. We provide a theoretical justification of this under suitable conditions. We demonstrate the accuracy, force computation efficiency, and robustness of our method on learning macroscopic closure models from a variety of microscopic systems, including those modeled by partial differential equations or molecular dynamics simulations.

Physical Models · Physics

Timothy Nest, Maxence Ernoult

Power efficiency is plateauing in the standard digital electronics realm such that new hardware, models, and algorithms are needed to reduce the costs of AI training. The combination of energy-based analog circuits and the Equilibrium Propagation (EP) algorithm constitutes a compelling alternative compute paradigm for gradient-based optimization of neural nets. Existing analog hardware accelerators, however, typically incorporate digital circuitry to sustain auxiliary non-weight-stationary operations, mitigate analog device imperfections, and leverage existing digital platforms. Such heterogeneous hardware lacks a supporting theoretical framework. In this work, we introduce \emph{Feedforward-tied Energy-based Models} (ff-EBMs), a hybrid model comprised of feedforward and energy-based blocks housed on digital and analog circuits. We derive a novel algorithm to compute gradients end-to-end in ff-EBMs by backpropagating and ``eq-propagating'' through feedforward and energy-based parts respectively, enabling EP to be applied flexibly on realistic architectures. We experimentally demonstrate the effectiveness of this approach on ff-EBMs using Deep Hopfield Networks (DHNs) as energy-based blocks, and show that a standard DHN can be arbitrarily split into any uniform size while maintaining or improving performance with increases in simulation speed of up to four times. We then train ff-EBMs on ImageNet32 where we establish a new state-of-the-art performance for the EP literature (46 top-1 \%). Our approach offers a principled, scalable, and incremental roadmap for the gradual integration of self-trainable analog computational primitives into existing digital accelerators.

Computer Vision

Linye Lyu, Jiawei Zhou, Daojing He, YU LI

Prior works on physical adversarial camouflage against vehicle detectors mainly focus on the effectiveness and robustness of the attack. The current most successful methods optimize 3D vehicle texture at a pixel level. However, this results in conspicuous and attention-grabbing patterns in the generated camouflage, which humans can easily identify. To address this issue, we propose a Customizable and Natural Camouflage Attack (CNCA) method by leveraging an off-the-shelf pre-trained diffusion model. By sampling the optimal texture image from the diffusion model with a user-specific text prompt, our method can generate natural and customizable adversarial camouflage while maintaining high attack performance. With extensive experiments on the digital and physical worlds and user studies, the results demonstrate that our proposed method can generate significantly more natural-looking camouflage than the state-of-the-art baselines while achieving competitive attack performance.

Physical Models

Michal Balcerak, Tamaz Amiranashvili, Andreas Wagner, Jonas Weidner, Petr Karnakov, Johannes C. Paetzold, Ivan Ezhov, Petros Koumoutsakos, Benedikt Wiestler, bjoern menze

Physical models in the form of partial differential equations serve as important priors for many under-constrained problems. One such application is tumor treatment planning, which relies on accurately estimating the spatial distribution of tumor cells within a patient’s anatomy. While medical imaging can detect the bulk of a tumor, it cannot capture the full extent of its spread, as low-concentration tumor cells often remain undetectable, particularly in glioblastoma, the most common primary brain tumor. Machine learning approaches struggle to estimate the complete tumor cell distribution due to a lack of appropriate training data. Consequently, most existing methods rely on physics-based simulations to generate anatomically and physiologically plausible estimations. However, these approaches face challenges with complex and unknown initial conditions and are constrained by overly rigid physical models. In this work, we introduce a novel method that integrates data-driven and physics-based cost functions, akin to Physics-Informed Neural Networks (PINNs). However, our approach parametrizes the solution directly on a dynamic discrete mesh, allowing for the effective modeling of complex biomechanical behaviors. Specifically, we propose a unique discretization scheme that quantifies how well the learned spatiotemporal distributions of tumor and brain tissues adhere to their respective growth and elasticity equations. This quantification acts as a regularization term, offering greater flexibility and improved integration of patient data compared to existing models. We demonstrate enhanced coverage of tumor recurrence areas using real-world data from a patient cohort, highlighting the potential of our method to improve model-driven treatment planning for glioblastoma in clinical practice.

ZAIXI ZHANG, Marinka Zitnik, Qi Liu

Designing ligand-binding proteins, such as enzymes and biosensors, is essential in bioengineering and protein biology. One critical step in this process involves designing protein pockets, the protein interface binding with the ligand. Current approaches to pocket generation often suffer from time-intensive physical computations or template-based methods, as well as compromised generation quality due to the overlooking of domain knowledge. To tackle these challenges, we propose PocketFlow, a generative model that incorporates protein-ligand interaction priors based on flow matching. During training, PocketFlow learns to model key types of protein-ligand interactions, such as hydrogen bonds. In the sampling, PocketFlow leverages multi-granularity guidance (overall binding affinity and interaction geometry constraints) to facilitate generating high-affinity and valid pockets. Extensive experiments show that PocketFlow outperforms baselines on multiple benchmarks, e.g., achieving an average improvement of 1.29 in Vina Score and 0.05 in scRMSD. Moreover, modeling interactions make PocketFlow a generalized generative model across multiple ligand modalities, including small molecules, peptides, and RNA.

Neuroscience, Cognitive Science

Luis Cubillos, Guy Revach, Matthew Mender, Joseph Costello, Hisham Temmar, Aren Hite, Diksha Anoop Kumar Zutshi, Dylan Wallace, Xiaoyong Ni, Madison Kelberman 等

People with brain or spinal cord-related paralysis often need to rely on others for basic tasks, limiting their independence. A potential solution is brain-machine interfaces (BMIs), which could allow them to voluntarily control external devices (e.g., robotic arm) by decoding brain activity to movement commands. In the past decade, deep-learning decoders have achieved state-of-the-art results in most BMI applications, ranging from speech production to finger control. However, the 'black-box' nature of deep-learning decoders could lead to unexpected behaviors, resulting in major safety concerns in real-world physical control scenarios. In these applications, explainable but lower-performing decoders, such as the Kalman filter (KF), remain the norm. In this study, we designed a BMI decoder based on KalmanNet, an extension of the KF that augments its operation with recurrent neural networks to compute the Kalman gain. This results in a varying “trust” that shifts between inputs and dynamics. We used this algorithm to predict finger movements from the brain activity of two monkeys. We compared KalmanNet results offline (pre-recorded data, $n=13$ days) and online (real-time predictions, $n=5$ days) with a simple KF and two recent deep-learning algorithms: tcFNN (non-ReFIT version) and LSTM. KalmanNet achieved comparable or better results than other deep learning models in offline and online modes, relying on the dynamical model for stopping while depending more on neural inputs for initiating movements. We further validated this mechanism by implementing a heteroscedastic KF that used the same strategy, and it also approached state-of-the-art performance while remaining in the explainable domain of standard KFs. However, we also see two downsides to KalmanNet. KalmanNet shares the limited generalization ability of existing deep-learning decoders, and its usage of the KF as an inductive bias limits its performance in the presence of unseen noise distributions. Despite this trade-off, our analysis successfully integrates traditional controls and modern deep-learning approaches to motivate high-performing yet still explainable BMI designs.

Physical Models · Physics

Timothée Devergne, Vladimir Kostic, Michele Parrinello, Massimiliano Pontil

We investigate learning the eigenfunctions of evolution operators for time-reversal invariant stochastic processes, a prime example being the Langevin equation used in molecular dynamics. Many physical or chemical processes described by this equation involve transitions between metastable states separated by high potential barriers that can hardly be crossed during a simulation. To overcome this bottleneck, data are collected via biased simulations that explore the state space more rapidly. We propose a framework for learning from biased simulations rooted in the infinitesimal generator of the process {and the associated resolvent operator}. We contrast our approach to more common ones based on the transfer operator, showing that it can provably learn the spectral properties of the unbiased system from biased data. In experiments, we highlight the advantages of our method over transfer operator approaches and recent developments based on generator learning, demonstrating its effectiveness in estimating eigenfunctions and eigenvalues. Importantly, we show that even with datasets containing only a few relevant transitions due to sub-optimal biasing, our approachrecovers relevant information about the transition mechanism.

Computer Vision

Junfeng Ni, Yixin Chen, Bohan Jing, Nan Jiang, Bin Wang, Bo Dai, Puhao Li, Yixin Zhu, Song-Chun Zhu, Siyuan Huang

We address the issue of physical implausibility in multi-view neural reconstruction. While implicit representations have gained popularity in multi-view 3D reconstruction, previous work struggles to yield physically plausible results, limiting their utility in domains requiring rigorous physical accuracy. This lack of plausibility stems from the absence of physics modeling in existing methods and their inability to recover intricate geometrical structures. In this paper, we introduce PHYRECON, the first approach to leverage both differentiable rendering and differentiable physics simulation to learn implicit surface representations. PHYRECON features a novel differentiable particle-based physical simulator built on neural implicit representations. Central to this design is an efficient transformation between SDF-based implicit representations and explicit surface points via our proposed Surface Points Marching Cubes (SP-MC), enabling differentiable learning with both rendering and physical losses. Additionally, PHYRECON models both rendering and physical uncertainty to identify and compensate for inconsistent and inaccurate monocular geometric priors. The physical uncertainty further facilitates physics-guided pixel sampling to enhance the learning of slender structures. By integrating these techniques, our model supports differentiable joint modeling of appearance, geometry, and physics. Extensive experiments demonstrate that PHYRECON significantly improves the reconstruction quality. Our results also exhibit superior physical stability in physical simulators, with at least a 40% improvement across all datasets, paving the way for future physics-based applications.

Physical Models · Physics

Yue Li, Yi Sun, Shida Sun, Juntian Ye, Yueyi Zhang, Feihu Xu, Zhiwei Xiong

Dynamic reconstruction in confocal non-line-of-sight imaging encounters great challenges since the dense raster-scanning manner limits the practical frame rate. A fewer pioneer works reconstruct high-resolution volumes from the under-scanning transient measurements but overlook temporal consistency among transient frames. To fully exploit multi-frame information, we propose the first spatial-temporal Mamba (ST-Mamba) based method tailored for dynamic reconstruction of transient videos. Our method capitalizes on neighbouring transient frames to aggregate the target 3D hidden volume. Specifically, the interleaved features extracted from the input transient frames are fed to the proposed ST-Mamba blocks, which leverage the time-resolving causality in transient measurement. The cross ST-Mamba blocks are then devised to integrate the adjacent transient features. The target high-resolution transient frame is subsequently recovered by the transient spreading module. After transient fusion and recovery, a physical-based network is employed to reconstruct the hidden volume. To tackle the substantial noise inherent in transient videos, we propose a wave-based loss function to impose constraints within the phasor field. Besides, we introduce a new dataset, comprising synthetic videos for training and real-world videos for evaluation. Extensive experiments showcase the superior performance of our method on both synthetic data and real world data captured by different imaging setups. The code and data are available at https://github.com/Depth2World/Dynamic_NLOS.

Physical Models · Physics

Takeshi Koshizuka, Masahiro Fujisawa, Yusuke Tanaka, Issei Sato

In this paper, we explores the expressivity and trainability of the Fourier Neural Operator (FNO). We establish a mean-field theory for the FNO, analyzing the behavior of the random FNO from an \emph{edge of chaos} perspective. Our investigation into the expressivity of a random FNO involves examining the ordered-chaos phase transition of the network based on the weight distribution. This phase transition demonstrates characteristics unique to the FNO, induced by mode truncation, while also showcasing similarities to those of densely connected networks. Furthermore, we identify a connection between expressivity and trainability: the ordered and chaotic phases correspond to regions of vanishing and exploding gradients, respectively. This finding provides a practical prerequisite for the stable training of the FNO. Our experimental results corroborate our theoretical findings.

Haoran He, Chenjia Bai, Ling Pan, Weinan Zhang, Bin Zhao, Xuelong Li

Learning a generalist embodied agent capable of completing multiple tasks poses challenges, primarily stemming from the scarcity of action-labeled robotic datasets. In contrast, a vast amount of human videos exist, capturing intricate tasks and interactions with the physical world. Promising prospects arise for utilizing actionless human videos for pre-training and transferring the knowledge to facilitate robot policy learning through limited robot demonstrations. However, it remains a challenge due to the domain gap between humans and robots. Moreover, it is difficult to extract useful information representing the dynamic world from human videos, because of its noisy and multimodal data structure. In this paper, we introduce a novel framework to tackle these challenges, which leverages a unified discrete diffusion to combine generative pre-training on human videos and policy fine-tuning on a small number of action-labeled robot videos. We start by compressing both human and robot videos into unified video tokens. In the pre-training stage, we employ a discrete diffusion model with a mask-and-replace diffusion strategy to predict future video tokens in the latent space. In the fine-tuning stage, we harness the imagined future videos to guide low-level action learning with a limited set of robot data. Experiments demonstrate that our method generates high-fidelity future videos for planning and enhances the fine-tuned policies compared to previous state-of-the-art approaches with superior performance.