To obtain fast solutions for governing physical equations in solid mechanics, we introduce a method that integrates the core ideas of the finite element method with physics-informed neural networks and concept of neural operators. This approach generalizes and enhances each method, learning the parametric solution for mechanical problems without relying on data from other resources (e.g. other numerical solvers). We propose directly utilizing the available discretized weak form in finite element packages to construct the loss functions algebraically, thereby demonstrating the ability to find solutions even in the presence of sharp discontinuities. Our focus is on micromechanics as an example, where knowledge of deformation and stress fields for a given heterogeneous microstructure is crucial for further design applications. The primary parameter under investigation is the Young's modulus distribution within the heterogeneous solid system. Our investigations reveal that physics-based training yields higher accuracy compared to purely data-driven approaches for unseen microstructures. Additionally, we offer two methods to directly improve the process of obtaining high-resolution solutions, avoiding the need to use basic interpolation techniques. First is based on an autoencoder approach to enhance the efficiency for calculation on high resolution grid point. Next, Fourier-based parametrization is utilized to address complex 2D and 3D problems in micromechanics. The latter idea aims to represent complex microstructures efficiently using Fourier coefficients. Comparisons with other well-known operator learning algorithms, further emphasize the advantages of the newly proposed method.
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Computational models of syntax are predominantly text-based. Here we propose that the most basic syntactic operations can be modeled directly from raw speech in a fully unsupervised way. We focus on one of the most ubiquitous and elementary properties of syntax -- concatenation. We introduce spontaneous concatenation: a phenomenon where convolutional neural networks (CNNs) trained on acoustic recordings of individual words start generating outputs with two or even three words concatenated without ever accessing data with multiple words in the input. We replicate this finding in several independently trained models with different hyperparameters and training data. Additionally, networks trained on two words learn to embed words into novel unobserved word combinations. To our knowledge, this is a previously unreported property of CNNs trained in the ciwGAN/fiwGAN setting on raw speech and has implications both for our understanding of how these architectures learn as well as for modeling syntax and its evolution from raw acoustic inputs.
In high-temperature plasma physics, a strong magnetic field is usually used to confine charged particles. Therefore, for studying the classical mathematical models of the physical problems it needs to consider the effect of external magnetic fields. One of the important model equations in plasma is the Vlasov-Poisson equation with an external magnetic field. This equation usually has multi-scale characteristics and rich physical properties, thus it is very important and meaningful to construct numerical methods that can maintain the physical properties inherited by the original systems over long time. This paper extends the corresponding theory in Cartesian coordinates to general orthogonal curvilinear coordinates, and proves that a Poisson-bracket structure can still be obtained after applying the corresponding finite element discretization. However, the Hamiltonian systems in the new coordinate systems generally cannot be decomposed into sub-systems that can be solved accurately, so it is impossible to use the splitting methods to construct the corresponding geometric integrators. Therefore, this paper proposes a semi-implicit method for strong magnetic fields and analyzes the asymptotic stability of this method.
This paper presents a fitted space-time finite element method for solving a parabolic advection-diffusion problem with a nonstationary interface. The jumping diffusion coefficient gives rise to the discontinuity of the spatial gradient of solution across the interface. We use the Banach-Necas-Babuska theorem to show the well-posedness of the continuous variational problem. A fully discrete finite-element based scheme is analyzed using the Galerkin method and unstructured fitted meshes. An optimal error estimate is established in a discrete energy norm under appropriate globally low but locally high regularity conditions. Some numerical results corroborate our theoretical results.
Drawing from the theory of stochastic differential equations, we introduce a novel sampling method for known distributions and a new algorithm for diffusion generative models with unknown distributions. Our approach is inspired by the concept of the reverse diffusion process, widely adopted in diffusion generative models. Additionally, we derive the explicit convergence rate based on the smooth ODE flow. For diffusion generative models and sampling, we establish a dimension-free particle approximation convergence result. Numerical experiments demonstrate the effectiveness of our method. Notably, unlike the traditional Langevin method, our sampling method does not require any regularity assumptions about the density function of the target distribution. Furthermore, we also apply our method to optimization problems.
We present a Bayesian method for multivariate changepoint detection that allows for simultaneous inference on the location of a changepoint and the coefficients of a logistic regression model for distinguishing pre-changepoint data from post-changepoint data. In contrast to many methods for multivariate changepoint detection, the proposed method is applicable to data of mixed type and avoids strict assumptions regarding the distribution of the data and the nature of the change. The regression coefficients provide an interpretable description of a potentially complex change. For posterior inference, the model admits a simple Gibbs sampling algorithm based on P\'olya-gamma data augmentation. We establish conditions under which the proposed method is guaranteed to recover the true underlying changepoint. As a testing ground for our method, we consider the problem of detecting topological changes in time series of images. We demonstrate that our proposed method $\mathtt{bclr}$, combined with a topological feature embedding, performs well on both simulated and real image data. The method also successfully recovers the location and nature of changes in more traditional changepoint tasks.
In this paper, we study an optimal control problem for a coupled non-linear system of reaction-diffusion equations with degenerate diffusion, consisting of two partial differential equations representing the density of cells and the concentration of the chemotactic agent. By controlling the concentration of the chemical substrates, this study can guide the optimal growth of cells. The novelty of this work lies on the direct and dual models that remain in a weak setting, which is uncommon in the recent literature for solving optimal control systems. Moreover, it is known that the adjoint problems offer a powerful approach to quantifying the uncertainty associated with model inputs. However, these systems typically lack closed-form solutions, making it challenging to obtain weak solutions. For that, the well-posedness of the direct problem is first well guaranteed. Then, the existence of an optimal control and the first-order optimality conditions are established. Finally, weak solutions for the adjoint system to the non-linear degenerate direct model, are introduced and investigated.
In this paper we discuss reduced order models for the approximation of parametric eigenvalue problems. In particular, we are interested in the presence of intersections or clusters of eigenvalues. The singularities originating by these phenomena make it hard a straightforward generalization of well known strategies normally used for standards PDEs. We investigate how the known results extend (or not) to higher order frequencies.
The novelty of the current work is precisely to propose a statistical procedure to combine estimates of the modal parameters provided by any set of Operational Modal Analysis (OMA) algorithms so as to avoid preference for a particular one and also to derive an approximate joint probability distribution of the modal parameters, from which engineering statistics of interest such as mean value and variance are readily provided. The effectiveness of the proposed strategy is assessed considering measured data from an actual centrifugal compressor. The statistics obtained for both forward and backward modal parameters are finally compared against modal parameters identified during standard stability verification testing (SVT) of centrifugal compressors prior to shipment, using classical Experimental Modal Analysis (EMA) algorithms. The current work demonstrates that combination of OMA algorithms can provide quite accurate estimates for both the modal parameters and the associated uncertainties with low computational costs.
In this work we propose a discretization of the second boundary condition for the Monge-Ampere equation arising in geometric optics and optimal transport. The discretization we propose is the natural generalization of the popular Oliker-Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.
One of the most promising applications of machine learning (ML) in computational physics is to accelerate the solution of partial differential equations (PDEs). The key objective of ML-based PDE solvers is to output a sufficiently accurate solution faster than standard numerical methods, which are used as a baseline comparison. We first perform a systematic review of the ML-for-PDE solving literature. Of articles that use ML to solve a fluid-related PDE and claim to outperform a standard numerical method, we determine that 79% (60/76) compare to a weak baseline. Second, we find evidence that reporting biases, especially outcome reporting bias and publication bias, are widespread. We conclude that ML-for-PDE solving research is overoptimistic: weak baselines lead to overly positive results, while reporting biases lead to underreporting of negative results. To a large extent, these issues appear to be caused by factors similar to those of past reproducibility crises: researcher degrees of freedom and a bias towards positive results. We call for bottom-up cultural changes to minimize biased reporting as well as top-down structural reforms intended to reduce perverse incentives for doing so.
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