Stress prediction in porous materials and structures is challenging due to the high computational cost associated with direct numerical simulations. Convolutional Neural Network (CNN) based architectures have recently been proposed as surrogates to approximate and extrapolate the solution of such multiscale simulations. These methodologies are usually limited to 2D problems due to the high computational cost of 3D voxel based CNNs. We propose a novel geometric learning approach based on a Graph Neural Network (GNN) that efficiently deals with three-dimensional problems by performing convolutions over 2D surfaces only. Following our previous developments using pixel-based CNN, we train the GNN to automatically add local fine-scale stress corrections to an inexpensively computed coarse stress prediction in the porous structure of interest. Our method is Bayesian and generates densities of stress fields, from which credible intervals may be extracted. As a second scientific contribution, we propose to improve the extrapolation ability of our network by deploying a strategy of online physics-based corrections. Specifically, we condition the posterior predictions of our probabilistic predictions to satisfy partial equilibrium at the microscale, at the inference stage. This is done using an Ensemble Kalman algorithm, to ensure tractability of the Bayesian conditioning operation. We show that this innovative methodology allows us to alleviate the effect of undesirable biases observed in the outputs of the uncorrected GNN, and improves the accuracy of the predictions in general.
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This paper presents the first application of the direct parametrisation method for invariant manifolds to a fully coupled multiphysics problem involving the nonlinear vibrations of deformable structures subjected to an electrostatic field. The formulation proposed is intended for model order reduction of electrostatically actuated resonating Micro-Electro-Mechanical Systems (MEMS). The continuous problem is first rewritten in a manner that can be directly handled by the parametrisation method, which relies upon automated asymptotic expansions. A new mixed fully Lagrangian formulation is thus proposed which contains only explicit polynomial nonlinearities, which is then discretised in the framework of finite element procedures. Validation is performed on the classical parallel plate configuration, where different formulations using either the general framework, or an approximation of the electrostatic field due to the geometric configuration selected, are compared. Reduced-order models along these formulations are also compared to full-order simulations operated with a time integration approach. Numerical results show a remarkable performance both in terms of accuracy and wealth of nonlinear effects that can be accounted for. In particular, the transition from hardening to softening behaviour of the primary resonance while increasing the constant voltage component of the electric actuation, is recovered. Secondary resonances leading to superharmonic and parametric resonances are also investigated with the reduced-order model.
Microring resonators (MRRs) are promising devices for time-delay photonic reservoir computing, but the impact of the different physical effects taking place in the MRRs on the reservoir computing performance is yet to be fully understood. We numerically analyze the impact of linear losses as well as thermo-optic and free-carrier effects relaxation times on the prediction error of the time-series task NARMA-10. We demonstrate the existence of three regions, defined by the input power and the frequency detuning between the optical source and the microring resonance, that reveal the cavity transition from linear to nonlinear regimes. One of these regions offers very low error in time-series prediction under relatively low input power and number of nodes while the other regions either lack nonlinearity or become unstable. This study provides insight into the design of the MRR and the optimization of its physical properties for improving the prediction performance of time-delay reservoir computing.
Many interesting physical problems described by systems of hyperbolic conservation laws are stiff, and thus impose a very small time-step because of the restrictive CFL stability condition. In this case, one can exploit the superior stability properties of implicit time integration which allows to choose the time-step only from accuracy requirements, and thus avoid the use of small time-steps. We discuss an efficient framework to devise high order implicit schemes for stiff hyperbolic systems without tailoring it to a specific problem. The nonlinearity of high order schemes, due to space- and time-limiting procedures which control nonphysical oscillations, makes the implicit time integration difficult, e.g.~because the discrete system is nonlinear also on linear problems. This nonlinearity of the scheme is circumvented as proposed in (Puppo et al., Comm.~Appl.~Math.~\& Comput., 2023) for scalar conservation laws, where a first order implicit predictor is computed to freeze the nonlinear coefficients of the essentially non-oscillatory space reconstruction, and also to assist limiting in time. In addition, we propose a novel conservative flux-centered a-posteriori time-limiting procedure using numerical entropy indicators to detect troubled cells. The numerical tests involve classical and artificially devised stiff problems using the Euler's system of gas-dynamics.
As in many fields of medical research, survival analysis has witnessed a growing interest in the application of deep learning techniques to model complex, high-dimensional, heterogeneous, incomplete, and censored medical data. Current methods often make assumptions about the relations between data that may not be valid in practice. In response, we introduce SAVAE (Survival Analysis Variational Autoencoder), a novel approach based on Variational Autoencoders. SAVAE contributes significantly to the field by introducing a tailored ELBO formulation for survival analysis, supporting various parametric distributions for covariates and survival time (as long as the log-likelihood is differentiable). It offers a general method that consistently performs well on various metrics, demonstrating robustness and stability through different experiments. Our proposal effectively estimates time-to-event, accounting for censoring, covariate interactions, and time-varying risk associations. We validate our model in diverse datasets, including genomic, clinical, and demographic data, with varying levels of censoring. This approach demonstrates competitive performance compared to state-of-the-art techniques, as assessed by the Concordance Index and the Integrated Brier Score. SAVAE also offers an interpretable model that parametrically models covariates and time. Moreover, its generative architecture facilitates further applications such as clustering, data imputation, and the generation of synthetic patient data through latent space inference from survival data.
Nowadays, machine learning algorithms continue to grow in complexity and require a substantial amount of computational resources and energy. For these reasons, there is a growing awareness of the development of new green algorithms and distributed AI can contribute to this. Federated learning (FL) is one of the most active research lines in machine learning, as it allows the training of collaborative models in a distributed way, an interesting option in many real-world environments, such as the Internet of Things, allowing the use of these models in edge computing devices. In this work, we present a FL method, based on a neural network without hidden layers, capable of generating a global collaborative model in a single training round, unlike traditional FL methods that require multiple rounds for convergence. This allows obtaining an effective and efficient model that simplifies the management of the training process. Moreover, this method preserve data privacy by design, a crucial aspect in current data protection regulations. We conducted experiments with large datasets and a large number of federated clients. Despite being based on a network model without hidden layers, it maintains in all cases competitive accuracy results compared to more complex state-of-the-art machine learning models. Furthermore, we show that the method performs equally well in both identically and non-identically distributed scenarios. Finally, it is an environmentally friendly algorithm as it allows significant energy savings during the training process compared to its centralized counterpart.
As a crossover frontier of physics and mechanics, quantum computing is showing its great potential in computational mechanics. However, quantum hardware noise remains a critical barrier to achieving accurate simulation results due to the limitation of the current hardware level. In this paper, we integrate error-mitigated quantum computing in data-driven computational mechanics, where the zero-noise extrapolation (ZNE) technique is employed to improve the accuracy of quantum computing. Numerical examples including multiscale simulation of a composite L-shaped beam are conducted with the quantum computer simulator Qpanda, and the results validate the effectiveness of the proposed method. We believe this work presents a promising step towards using the power of quantum computing in computational mechanics.
Cubic-quartic regularization models for solving polynomial subproblems in third-order tensor methods
High-order tensor methods for solving both convex and nonconvex optimization problems have generated significant research interest, leading to algorithms with optimal global rates of convergence and local rates that are faster than Newton's method. On each iteration, these methods require the unconstrained local minimization of a (potentially nonconvex) multivariate polynomial of degree higher than two, constructed using third-order (or higher) derivative information, and regularized by an appropriate power of regularization. Developing efficient techniques for solving such subproblems is an ongoing topic of research, and this paper addresses the case of the third-order tensor subproblem. We propose the CQR algorithmic framework, for minimizing a nonconvex Cubic multivariate polynomial with Quartic Regularisation, by minimizing a sequence of local quadratic models that incorporate simple cubic and quartic terms. The role of the cubic term is to crudely approximate local tensor information, while the quartic one controls model regularization and progress. We provide necessary and sufficient optimality conditions that fully characterise the global minimizers of these cubic-quartic models. We then turn these conditions into secular equations that can be solved using nonlinear eigenvalue techniques. We show, using our optimality characterisations, that a CQR algorithmic variant has the optimal-order evaluation complexity of $\mathcal{O}(\epsilon^{-3/2})$ when applied to minimizing our quartically-regularised cubic subproblem, which can be further improved in special cases. We propose practical CQR variants that use local tensor information to construct the local cubic-quartic models. We test these variants numerically and observe them to be competitive with ARC and other subproblem solvers on typical instances and even superior on ill-conditioned subproblems with special structure.
Partitioned neural network functions are used to approximate the solution of partial differential equations. The problem domain is partitioned into non-overlapping subdomains and the partitioned neural network functions are defined on the given non-overlapping subdomains. Each neural network function then approximates the solution in each subdomain. To obtain the convergent neural network solution, certain continuity conditions on the partitioned neural network functions across the subdomain interface need to be included in the loss function, that is used to train the parameters in the neural network functions. In our work, by introducing suitable interface values, the loss function is reformulated into a sum of localized loss functions and each localized loss function is used to train the corresponding local neural network parameters. In addition, to accelerate the neural network solution convergence, the localized loss function is enriched with an augmented Lagrangian term, where the interface condition and the boundary condition are enforced as constraints on the local solutions by using Lagrange multipliers. The local neural network parameters and Lagrange multipliers are then found by optimizing the localized loss function. To take the advantage of the localized loss function for the parallel computation, an iterative algorithm is also proposed. For the proposed algorithms, their training performance and convergence are numerically studied for various test examples.
We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.
We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.
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