We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learn the transfer operator of the system, that in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the context of transfer operator regression. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.
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In modern computational materials science, deep learning has shown the capability to predict interatomic potentials, thereby supporting and accelerating conventional simulations. However, existing models typically sacrifice either accuracy or efficiency. Moreover, lightweight models are highly demanded for offering simulating systems on a considerably larger scale at reduced computational costs. A century ago, Felix Bloch demonstrated how leveraging the equivariance of the translation operation on a crystal lattice (with geometric symmetry) could significantly reduce the computational cost of determining wavefunctions and accurately calculate material properties. Here, we introduce a lightweight equivariant interaction graph neural network (LEIGNN) that can enable accurate and efficient interatomic potential and force predictions in crystals. Rather than relying on higher-order representations, LEIGNN employs a scalar-vector dual representation to encode equivariant features. By extracting both local and global structures from vector representations and learning geometric symmetry information, our model remains lightweight while ensuring prediction accuracy and robustness through the equivariance. Our results show that LEIGNN consistently outperforms the prediction performance of the representative baselines and achieves significant efficiency across diverse datasets, which include catalysts, molecules, and organic isomers. Finally, to further validate the predicted interatomic potentials from our model, we conduct classical molecular dynamics (MD) and ab initio MD simulation across various systems, including solid, liquid, and gas. It is found that LEIGNN can achieve the accuracy of ab initio MD and retain the computational efficiency of classical MD across all examined systems, demonstrating its accuracy, efficiency, and universality.
Generalized linear models (GLMs) are popular for data-analysis in almost all quantitative sciences, but the choice of likelihood family and link function is often difficult. This motivates the search for likelihoods and links that minimize the impact of potential misspecification. We perform a large-scale simulation study on double-bounded and lower-bounded response data where we systematically vary both true and assumed likelihoods and links. In contrast to previous studies, we also study posterior calibration and uncertainty metrics in addition to point-estimate accuracy. Our results indicate that certain likelihoods and links can be remarkably robust to misspecification, performing almost on par with their respective true counterparts. Additionally, normal likelihood models with identity link (i.e., linear regression) often achieve calibration comparable to the more structurally faithful alternatives, at least in the studied scenarios. On the basis of our findings, we provide practical suggestions for robust likelihood and link choices in GLMs.
The maximum likelihood estimator (MLE) is pivotal in statistical inference, yet its application is often hindered by the absence of closed-form solutions for many models. This poses challenges in real-time computation scenarios, particularly within embedded systems technology, where numerical methods are impractical. This study introduces a generalized form of the MLE that yields closed-form estimators under certain conditions. We derive the asymptotic properties of the proposed estimator and demonstrate that our approach retains key properties such as invariance under one-to-one transformations, strong consistency, and an asymptotic normal distribution. The effectiveness of the generalized MLE is exemplified through its application to the Gamma, Nakagami, and Beta distributions, showcasing improvements over the traditional MLE. Additionally, we extend this methodology to a bivariate gamma distribution, successfully deriving closed-form estimators. This advancement presents significant implications for real-time statistical analysis across various applications.
This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures on a compact and convex subset of $\mathbb{R}^d$, metrized with the Wasserstein-2 distance $W$. We begin by introducing a natural construction of submanifolds $\Lambda$ of probability measures equipped with metric $W_\Lambda$, the geodesic restriction of $W$ to $\Lambda$. In contrast to other constructions, these submanifolds are not necessarily flat, but still allow for local linearizations in a similar fashion to Riemannian submanifolds of $\mathbb{R}^d$. We then show how the latent manifold structure of $(\Lambda,W_{\Lambda})$ can be learned from samples $\{\lambda_i\}_{i=1}^N$ of $\Lambda$ and pairwise extrinsic Wasserstein distances $W$ only. In particular, we show that the metric space $(\Lambda,W_{\Lambda})$ can be asymptotically recovered in the sense of Gromov--Wasserstein from a graph with nodes $\{\lambda_i\}_{i=1}^N$ and edge weights $W(\lambda_i,\lambda_j)$. In addition, we demonstrate how the tangent space at a sample $\lambda$ can be asymptotically recovered via spectral analysis of a suitable "covariance operator" using optimal transport maps from $\lambda$ to sufficiently close and diverse samples $\{\lambda_i\}_{i=1}^N$. The paper closes with some explicit constructions of submanifolds $\Lambda$ and numerical examples on the recovery of tangent spaces through spectral analysis.
Selection models are ubiquitous in statistics. In recent years, they have regained considerable popularity as the working inferential models in many selective inference problems. In this paper, we derive an asymptotic expansion of the local likelihood ratios of selection models. We show that under mild regularity conditions, they are asymptotically equivalent to a sequence of Gaussian selection models. This generalizes the Local Asymptotic Normality framework of Le Cam (1960). Furthermore, we derive the asymptotic shape of Bayesian posterior distributions constructed from selection models, and show that they can be significantly miscalibrated in a frequentist sense.
We present a formulation for high-order generalized periodicity conditions in the context of a high-order electromechanical theory including flexoelectricity, strain gradient elasticity and gradient dielectricity, with the goal of studying periodic architected metamaterials. Such theory results in fourth-order governing partial differential equations, and the periodicity conditions involve continuity across the periodic boundary of primal fields (displacement and electric potential) and their normal derivatives, continuity of the corresponding dual generalized forces (tractions, double tractions, surface charge density and double surface charge density). Rather than imposing these conditions numerically as explicit constraints, we develop an approximation space which fulfils generalized periodicity by construction. Our method naturally allows us to impose general macroscopic fields (strains/stresses and electric fields/electric displacements) along arbitrary directions, enabling the characterization of the material anisotropy. We apply the proposed method to study periodic architected metamaterials with apparent piezoelectricity. We first verify the method by directly comparing the results with a large periodic structure, then apply it to evaluate the anisotropic apparently piezoelectricity of a geometrically polarized 2D lattice, and finally demonstrate the application of the method in a 3D architected metamaterial.
In harsh environments, organisms may self-organize into spatially patterned systems in various ways. So far, studies of ecosystem spatial self-organization have primarily focused on apparent orders reflected by regular patterns. However, self-organized ecosystems may also have cryptic orders that can be unveiled only through certain quantitative analyses. Here we show that disordered hyperuniformity as a striking class of hidden orders can exist in spatially self-organized vegetation landscapes. By analyzing the high-resolution remotely sensed images across the American drylands, we demonstrate that it is not uncommon to find disordered hyperuniform vegetation states characterized by suppressed density fluctuations at long range. Such long-range hyperuniformity has been documented in a wide range of microscopic systems. Our finding contributes to expanding this domain to accommodate natural landscape ecological systems. We use theoretical modeling to propose that disordered hyperuniform vegetation patterning can arise from three generalized mechanisms prevalent in dryland ecosystems, including (1) critical absorbing states driven by an ecological legacy effect, (2) scale-dependent feedbacks driven by plant-plant facilitation and competition, and (3) density-dependent aggregation driven by plant-sediment feedbacks. Our modeling results also show that disordered hyperuniform patterns can help ecosystems cope with arid conditions with enhanced functioning of soil moisture acquisition. However, this advantage may come at the cost of slower recovery of ecosystem structure upon perturbations. Our work highlights that disordered hyperuniformity as a distinguishable but underexplored ecosystem self-organization state merits systematic studies to better understand its underlying mechanisms, functioning, and resilience.
The reduction of Hamiltonian systems aims to build smaller reduced models, valid over a certain range of time and parameters, in order to reduce computing time. By maintaining the Hamiltonian structure in the reduced model, certain long-term stability properties can be preserved. In this paper, we propose a non-linear reduction method for models coming from the spatial discretization of partial differential equations: it is based on convolutional auto-encoders and Hamiltonian neural networks. Their training is coupled in order to simultaneously learn the encoder-decoder operators and the reduced dynamics. Several test cases on non-linear wave dynamics show that the method has better reduction properties than standard linear Hamiltonian reduction methods.
Nowadays, deep-learning image coding solutions have shown similar or better compression efficiency than conventional solutions based on hand-crafted transforms and spatial prediction techniques. These deep-learning codecs require a large training set of images and a training methodology to obtain a suitable model (set of parameters) for efficient compression. The training is performed with an optimization algorithm which provides a way to minimize the loss function. Therefore, the loss function plays a key role in the overall performance and includes a differentiable quality metric that attempts to mimic human perception. The main objective of this paper is to study the perceptual impact of several image quality metrics that can be used in the loss function of the training process, through a crowdsourcing subjective image quality assessment study. From this study, it is possible to conclude that the choice of the quality metric is critical for the perceptual performance of the deep-learning codec and that can vary depending on the image content.
We present a framework for the efficient computation of optimal Bayesian decisions under intractable likelihoods, by learning a surrogate model for the expected utility (or its distribution) as a function of the action and data spaces. We leverage recent advances in simulation-based inference and Bayesian optimization to develop active learning schemes to choose where in parameter and action spaces to simulate. This allows us to learn the optimal action in as few simulations as possible. The resulting framework is extremely simulation efficient, typically requiring fewer model calls than the associated posterior inference task alone, and a factor of $100-1000$ more efficient than Monte-Carlo based methods. Our framework opens up new capabilities for performing Bayesian decision making, particularly in the previously challenging regime where likelihoods are intractable, and simulations expensive.
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