Conventional neural network elastoplasticity models are often perceived as lacking interpretability. This paper introduces a two-step machine learning approach that returns mathematical models interpretable by human experts. In particular, we introduce a surrogate model where yield surfaces are expressed in terms of a set of single-variable feature mappings obtained from supervised learning. A postprocessing step is then used to re-interpret the set of single-variable neural network mapping functions into mathematical form through symbolic regression. This divide-and-conquer approach provides several important advantages. First, it enables us to overcome the scaling issue of symbolic regression algorithms. From a practical perspective, it enhances the portability of learned models for partial differential equation solvers written in different programming languages. Finally, it enables us to have a concrete understanding of the attributes of the materials, such as convexity and symmetries of models, through automated derivations and reasoning. Numerical examples have been provided, along with an open-source code to enable third party validation.
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ACM/IEEE第23屆模型驅動工程語言和系統國際會議，是模型驅動軟件和系統工程的首要會議系列，由ACM-SIGSOFT和IEEE-TCSE支持組織。自1998年以來，模型涵蓋了建模的各個方面，從語言和方法到工具和應用程序。模特的參加者來自不同的背景，包括研究人員、學者、工程師和工業專業人士。MODELS 2019是一個論壇，參與者可以圍繞建模和模型驅動的軟件和系統交流前沿研究成果和創新實踐經驗。今年的版本將為建模社區提供進一步推進建模基礎的機會，并在網絡物理系統、嵌入式系統、社會技術系統、云計算、大數據、機器學習、安全、開源等新興領域提出建模的創新應用以及可持續性。
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We study Bayesian methods for large-scale linear inverse problems, focusing on the challenging task of hyperparameter estimation. Typical hierarchical Bayesian formulations that follow a Markov Chain Monte Carlo approach are possible for small problems with very few hyperparameters but are not computationally feasible for problems with a very large number of unknown parameters. In this work, we describe an empirical Bayesian (EB) method to estimate hyperparameters that maximize the marginal posterior, i.e., the probability density of the hyperparameters conditioned on the data, and then we use the estimated values to compute the posterior of the inverse parameters. For problems where the computation of the square root and inverse of prior covariance matrices are not feasible, we describe an approach based on the generalized Golub-Kahan bidiagonalization to approximate the marginal posterior and seek hyperparameters that minimize the approximate marginal posterior. Numerical results from seismic and atmospheric tomography demonstrate the accuracy, robustness, and potential benefits of the proposed approach.
Sequential neural posterior estimation (SNPE) techniques have been recently proposed for dealing with simulation-based models with intractable likelihoods. Unlike approximate Bayesian computation, SNPE techniques learn the posterior from sequential simulation using neural network-based conditional density estimators by minimizing a specific loss function. The SNPE method proposed by Lueckmann et al. (2017) used a calibration kernel to boost the sample weights around the observed data, resulting in a concentrated loss function. However, the use of calibration kernels may increase the variances of both the empirical loss and its gradient, making the training inefficient. To improve the stability of SNPE, this paper proposes to use an adaptive calibration kernel and several variance reduction techniques. The proposed method greatly speeds up the process of training, and provides a better approximation of the posterior than the original SNPE method and some existing competitors as confirmed by numerical experiments.
In this paper, we propose a robust low-order stabilization-free virtual element method on quadrilateral meshes for linear elasticity that is based on the stress-hybrid principle. We refer to this approach as the Stress-Hybrid Virtual Element Method (SH-VEM). In this method, the Hellinger$-$Reissner variational principle is adopted, wherein both the equilibrium equations and the strain-displacement relations are variationally enforced. We consider small-strain deformations of linear elastic solids in the compressible and near-incompressible regimes over quadrilateral (convex and nonconvex) meshes. Within an element, the displacement field is approximated as a linear combination of canonical shape functions that are $\textit{virtual}$. The stress field, similar to the stress-hybrid finite element method of Pian and Sumihara, is represented using a linear combination of symmetric tensor polynomials. A 5-parameter expansion of the stress field is used in each element, with stress transformation equations applied on distorted quadrilaterals. In the variational statement of the strain-displacement relations, the divergence theorem is invoked to express the stress coefficients in terms of the nodal displacements. This results in a formulation with solely the nodal displacements as unknowns. Numerical results are presented for several benchmark problems from linear elasticity. We show that SH-VEM is free of volumetric and shear locking, and it converges optimally in the $L^2$ norm and energy seminorm of the displacement field, and in the $L^2$ norm of the hydrostatic stress.
Nonlinear systems arising from time integrators like Backward Euler can sometimes be reformulated as optimization problems, known as incremental potentials. We show through a comprehensive experimental analysis that the widely used Projected Newton method, which relies on unconditional semidefinite projection of Hessian contributions, typically exhibits a reduced convergence rate compared to classical Newton's method. We demonstrate how factors like resolution, element order, projection method, material model and boundary handling impact convergence of Projected Newton and Newton. Drawing on these findings, we propose the hybrid method Project-on-Demand Newton, which projects only conditionally, and show that it enjoys both the robustness of Projected Newton and convergence rate of Newton. We additionally introduce Kinetic Newton, a regularization-based method that takes advantage of the structure of incremental potentials and avoids projection altogether. We compare the four solvers on hyperelasticity and contact problems. We also present a nuanced discussion of convergence criteria, and propose a new acceleration-based criterion that avoids problems associated with existing residual norm criteria and is easier to interpret. We finally address a fundamental limitation of the Armijo backtracking line search that occasionally blocks convergence, especially for stiff problems. We propose a novel parameter-free, robust line search technique to eliminate this issue.
When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer a significant advantage in dealing with complex geometries, eliminating the need for generating unstructured body-fitted meshes. However, current unfitted finite elements on nonlinear geometries are restricted to implicit (possibly high-order) level set geometries. In this work, we introduce a novel automatic computational pipeline to approximate solutions of partial differential equations on domains defined by explicit nonlinear boundary representations. For the geometrical discretization, we propose a novel algorithm to generate quadratures for the bulk and surface integration on nonlinear polytopes required to compute all the terms in unfitted finite element methods. The algorithm relies on a nonlinear triangulation of the boundary, a kd-tree refinement of the surface cells that simplify the nonlinear intersections of surface and background cells to simple cases that are diffeomorphically equivalent to linear intersections, robust polynomial root-finding algorithms and surface parameterization techniques. We prove the correctness of the proposed algorithm. We have successfully applied this algorithm to simulate partial differential equations with unfitted finite elements on nonlinear domains described by computer-aided design models, demonstrating the robustness of the geometric algorithm and showing high-order accuracy of the overall method.
We address the problem of testing conditional mean and conditional variance for non-stationary data. We build e-values and p-values for four types of non-parametric composite hypotheses with specified mean and variance as well as other conditions on the shape of the data-generating distribution. These shape conditions include symmetry, unimodality, and their combination. Using the obtained e-values and p-values, we construct tests via e-processes, also known as testing by betting, as well as some tests based on combining p-values for comparison. Although we mainly focus on one-sided tests, the two-sided test for the mean is also studied. Simulation and empirical studies are conducted under a few settings, and they illustrate features of the methods based on e-processes.
Refinement calculus provides a structured framework for the progressive and modular development of programs, ensuring their correctness throughout the refinement process. This paper introduces a refinement calculus tailored for quantum programs. To this end, we first study the partial correctness of nondeterministic programs within a quantum while language featuring prescription statements. Orthogonal projectors, which are equivalent to subspaces of the state Hilbert space, are taken as assertions for quantum states. In addition to the denotational semantics where a nondeterministic program is associated with a set of trace-nonincreasing super-operators, we also present their semantics in transforming a postcondition to the weakest liberal postconditions and, conversely, transforming a precondition to the strongest postconditions. Subsequently, refinement rules are introduced based on these dual semantics, offering a systematic approach to the incremental development of quantum programs applicable in various contexts. To illustrate the practical application of the refinement calculus, we examine examples such as the implementation of a $Z$-rotation gate, the repetition code, and the quantum-to-quantum Bernoulli factory. Furthermore, we present Quire, a Python-based interactive prototype tool that provides practical support to programmers engaged in the stepwise development of correct quantum programs.
We propose and compare methods for the analysis of extreme events in complex systems governed by PDEs that involve random parameters, in situations where we are interested in quantifying the probability that a scalar function of the system's solution is above a threshold. If the threshold is large, this probability is small and its accurate estimation is challenging. To tackle this difficulty, we blend theoretical results from large deviation theory (LDT) with numerical tools from PDE-constrained optimization. Our methods first compute parameters that minimize the LDT-rate function over the set of parameters leading to extreme events, using adjoint methods to compute the gradient of this rate function. The minimizers give information about the mechanism of the extreme events as well as estimates of their probability. We then propose a series of methods to refine these estimates, either via importance sampling or geometric approximation of the extreme event sets. Results are formulated for general parameter distributions and detailed expressions are provided when Gaussian distributions. We give theoretical and numerical arguments showing that the performance of our methods is insensitive to the extremeness of the events we are interested in. We illustrate the application of our approach to quantify the probability of extreme tsunami events on shore. Tsunamis are typically caused by a sudden, unpredictable change of the ocean floor elevation during an earthquake. We model this change as a random process, which takes into account the underlying physics. We use the one-dimensional shallow water equation to model tsunamis numerically. In the context of this example, we present a comparison of our methods for extreme event probability estimation, and find which type of ocean floor elevation change leads to the largest tsunamis on shore.
Objective: Clinical deep phenotyping and phenotype annotation play a critical role in both the diagnosis of patients with rare disorders as well as in building computationally-tractable knowledge in the rare disorders field. These processes rely on using ontology concepts, often from the Human Phenotype Ontology, in conjunction with a phenotype concept recognition task (supported usually by machine learning methods) to curate patient profiles or existing scientific literature. With the significant shift in the use of large language models (LLMs) for most NLP tasks, we examine the performance of the latest Generative Pre-trained Transformer (GPT) models underpinning ChatGPT as a foundation for the tasks of clinical phenotyping and phenotype annotation. Materials and Methods: The experimental setup of the study included seven prompts of various levels of specificity, two GPT models (gpt-3.5-turbo and gpt-4.0) and two established gold standard corpora for phenotype recognition, one consisting of publication abstracts and the other clinical observations. Results: Our results show that, with an appropriate setup, these models can achieve state of the art performance. The best run, using few-shot learning, achieved 0.58 macro F1 score on publication abstracts and 0.75 macro F1 score on clinical observations, the former being comparable with the state of the art, while the latter surpassing the current best in class tool. Conclusion: While the results are promising, the non-deterministic nature of the outcomes, the high cost and the lack of concordance between different runs using the same prompt and input make the use of these LLMs challenging for this particular task.
In this paper, we introduce an Abaqus UMAT subroutine for a family of constitutive models for the viscoelastic response of isotropic elastomers of any compressibility -- including fully incompressible elastomers -- undergoing finite deformations. The models can be chosen to account for a wide range of non-Gaussian elasticities, as well as for a wide range of nonlinear viscosities. From a mathematical point of view, the structure of the models is such that the viscous dissipation is characterized by an internal variable $\textbf{C}^v$, subject to the physically-based constraint $\det\textbf{C}^v=1$, that is solution of a nonlinear first-order ODE in time. This ODE is solved by means of an explicit Runge-Kutta scheme of high order capable of preserving the constraint $\det\textbf{C}^v=1$ identically. The accuracy and convergence of the code is demonstrated numerically by comparison with an exact solution for several of the Abaqus built-in hybrid finite elements, including the simplicial elements C3D4H and C3D10H and the hexahedral elements C3D8H and C3D20H. The last part of this paper is devoted to showcasing the capabilities of the code by deploying it to compute the homogenized response of a bicontinuous rubber blend.
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