In the present work, we introduce a novel approach to enhance the precision of reduced order models by exploiting a multi-fidelity perspective and DeepONets. Reduced models provide a real-time numerical approximation by simplifying the original model. The error introduced by the such operation is usually neglected and sacrificed in order to reach a fast computation. We propose to couple the model reduction to a machine learning residual learning, such that the above-mentioned error can be learned by a neural network and inferred for new predictions. We emphasize that the framework maximizes the exploitation of high-fidelity information, using it for building the reduced order model and for learning the residual. In this work, we explore the integration of proper orthogonal decomposition (POD), and gappy POD for sensors data, with the recent DeepONet architecture. Numerical investigations for a parametric benchmark function and a nonlinear parametric Navier-Stokes problem are presented.
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Deep learning techniques depend on large datasets whose annotation is time-consuming. To reduce annotation burden, the self-training (ST) and active-learning (AL) methods have been developed as well as methods that combine them in an iterative fashion. However, it remains unclear when each method is the most useful, and when it is advantageous to combine them. In this paper, we propose a new method that combines ST with AL using Test-Time Augmentations (TTA). First, TTA is performed on an initial teacher network. Then, cases for annotation are selected based on the lowest estimated Dice score. Cases with high estimated scores are used as soft pseudo-labels for ST. The selected annotated cases are trained with existing annotated cases and ST cases with border slices annotations. We demonstrate the method on MRI fetal body and placenta segmentation tasks with different data variability characteristics. Our results indicate that ST is highly effective for both tasks, boosting performance for in-distribution (ID) and out-of-distribution (OOD) data. However, while self-training improved the performance of single-sequence fetal body segmentation when combined with AL, it slightly deteriorated performance of multi-sequence placenta segmentation on ID data. AL was helpful for the high variability placenta data, but did not improve upon random selection for the single-sequence body data. For fetal body segmentation sequence transfer, combining AL with ST following ST iteration yielded a Dice of 0.961 with only 6 original scans and 2 new sequence scans. Results using only 15 high-variability placenta cases were similar to those using 50 cases. Code is available at: //github.com/Bella31/TTA-quality-estimation-ST-AL
The distributed task allocation problem, as one of the most interesting distributed optimization challenges, has received considerable research attention recently. Previous works mainly focused on the task allocation problem in a population of individuals, where there are no constraints for affording task amounts. The latter condition, however, cannot always be hold. In this paper, we study the task allocation problem with constraints of task allocation in a game-theoretical framework. We assume that each individual can afford different amounts of task and the cost function is convex. To investigate the problem in the framework of population games, we construct a potential game and calculate the fitness function for each individual. We prove that when the Nash equilibrium point in the potential game is in the feasible solutions for the limited task allocation problem, the Nash equilibrium point is the unique globally optimal solution. Otherwise, we also derive analytically the unique globally optimal solution. In addition, in order to confirm our theoretical results, we consider the exponential and quadratic forms of cost function for each agent. Two algorithms with the mentioned representative cost functions are proposed to numerically seek the optimal solution to the limited task problems. We further perform Monte Carlo simulations which provide agreeing results with our analytical calculations.
In this paper we develop a numerical method for efficiently approximating solutions of certain Zakai equations in high dimensions. The key idea is to transform a given Zakai SPDE into a PDE with random coefficients. We show that under suitable regularity assumptions on the coefficients of the Zakai equation, the corresponding random PDE admits a solution random field which, for almost all realizations of the random coefficients, can be written as a classical solution of a linear parabolic PDE. This makes it possible to apply the Feynman--Kac formula to obtain an efficient Monte Carlo scheme for computing approximate solutions of Zakai equations. The approach achieves good results in up to 25 dimensions with fast run times.
In the present paper, we develop a new goodness-of-fit test for the Birnbaum- Saunders distribution based on the probability plot. We utilize the sample correlation coefficient from the Birnbaum-Saunders probability plot as a measure of goodness of fit. Unfortunately, it is impossible or extremely difficult to obtain an explicit distribution of this sample correlation coefficient. To address this challenge, we employ extensive Monte Carlo simulations to obtain the empirical distribution of the sample correlation coefficient from the Birnbaum-Saunders probability plot. This empirical distribution allows us to determine the critical values alongside their corresponding significance levels, thus facilitating the computation of the p-value when the sample correlation coefficient is obtained. Finally, two real-data examples are provided for illustrative purposes.
Many inductive logic programming approaches struggle to learn programs from noisy data. To overcome this limitation, we introduce an approach that learns minimal description length programs from noisy data, including recursive programs. Our experiments on several domains, including drug design, game playing, and program synthesis, show that our approach can outperform existing approaches in terms of predictive accuracies and scale to moderate amounts of noise.
Quantization summarizes continuous distributions by calculating a discrete approximation. Among the widely adopted methods for data quantization is Lloyd's algorithm, which partitions the space into Vorono\"i cells, that can be seen as clusters, and constructs a discrete distribution based on their centroids and probabilistic masses. Lloyd's algorithm estimates the optimal centroids in a minimal expected distance sense, but this approach poses significant challenges in scenarios where data evaluation is costly, and relates to rare events. Then, the single cluster associated to no event takes the majority of the probability mass. In this context, a metamodel is required and adapted sampling methods are necessary to increase the precision of the computations on the rare clusters.
A central problem in computational statistics is to convert a procedure for sampling combinatorial from an objects into a procedure for counting those objects, and vice versa. Weconsider sampling problems coming from *Gibbs distributions*, which are probability distributions of the form $\mu^\Omega_\beta(\omega) \propto e^{\beta H(\omega)}$ for $\beta$ in an interval $[\beta_\min, \beta_\max]$ and $H( \omega ) \in \{0 \} \cup [1, n]$. The *partition function* is the normalization factor $Z(\beta)=\sum_{\omega \in\Omega}e^{\beta H(\omega)}$. Two important parameters are the log partition ratio $q = \log \tfrac{Z(\beta_\max)}{Z(\beta_\min)}$ and the vector of counts $c_x = |H^{-1}(x)|$. Our first result is an algorithm to estimate the counts $c_x$ using roughly $\tilde O( \frac{q}{\epsilon^2})$ samples for general Gibbs distributions and $\tilde O( \frac{n^2}{\epsilon^2} )$ samples for integer-valued distributions (ignoring some second-order terms and parameters). We show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs and perfect matchings in a graph. We develop a key subroutine for global estimation of the partition function. Specifically, we produce a data structure to estimate $Z(\beta)$ for \emph{all} values $\beta$, without further samples. Constructing the data structure requires $O(\frac{q \log n}{\epsilon^2})$ samples for general Gibbs distributions and $O(\frac{n^2 \log n}{\epsilon^2} + n \log q)$ samples for integer-valued distributions. This improves over a prior algorithm of Kolmogorov (2018) which computes the single point estimate $Z(\beta_\max)$ using $\tilde O(\frac{q}{\epsilon^2})$ samples. We also show that this complexity is optimal as a function of $n$ and $q$ up to logarithmic terms.
We analyze the dynamics of streaming stochastic gradient descent (SGD) in the high-dimensional limit when applied to generalized linear models and multi-index models (e.g. logistic regression, phase retrieval) with general data-covariance. In particular, we demonstrate a deterministic equivalent of SGD in the form of a system of ordinary differential equations that describes a wide class of statistics, such as the risk and other measures of sub-optimality. This equivalence holds with overwhelming probability when the model parameter count grows proportionally to the number of data. This framework allows us to obtain learning rate thresholds for stability of SGD as well as convergence guarantees. In addition to the deterministic equivalent, we introduce an SDE with a simplified diffusion coefficient (homogenized SGD) which allows us to analyze the dynamics of general statistics of SGD iterates. Finally, we illustrate this theory on some standard examples and show numerical simulations which give an excellent match to the theory.
Time-dependent basis reduced order models (TDB ROMs) have successfully been used for approximating the solution to nonlinear stochastic partial differential equations (PDEs). For many practical problems of interest, discretizing these PDEs results in massive matrix differential equations (MDEs) that are too expensive to solve using conventional methods. While TDB ROMs have the potential to significantly reduce this computational burden, they still suffer from the following challenges: (i) inefficient for general nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the presence of small singular values, and (iv) error accumulation due to fixed rank. To this end, we present a scalable method based on oblique projections for solving TDB ROMs that is computationally efficient, minimally intrusive, robust in the presence of small singular values, rank-adaptive, and highly parallelizable. These favorable properties are achieved via low-rank approximation of the time discrete MDE. Using the discrete empirical interpolation method (DEIM), a low-rank decomposition is computed at each iteration of the time stepping scheme, enabling a near-optimal approximation at a fraction of the cost. We coin the new approach TDB-CUR since it is equivalent to a CUR decomposition based on sparse row and column samples of the MDE. We also propose a rank-adaptive procedure to control the error on-the-fly. Numerical results demonstrate the accuracy, efficiency, and robustness of the new method for a diverse set of problems.
Long-span bridges are subjected to a multitude of dynamic excitations during their lifespan. To account for their effects on the structural system, several load models are used during design to simulate the conditions the structure is likely to experience. These models are based on different simplifying assumptions and are generally guided by parameters that are stochastically identified from measurement data, making their outputs inherently uncertain. This paper presents a probabilistic physics-informed machine-learning framework based on Gaussian process regression for reconstructing dynamic forces based on measured deflections, velocities, or accelerations. The model can work with incomplete and contaminated data and offers a natural regularization approach to account for noise in the measurement system. An application of the developed framework is given by an aerodynamic analysis of the Great Belt East Bridge. The aerodynamic response is calculated numerically based on the quasi-steady model, and the underlying forces are reconstructed using sparse and noisy measurements. Results indicate a good agreement between the applied and the predicted dynamic load and can be extended to calculate global responses and the resulting internal forces. Uses of the developed framework include validation of design models and assumptions, as well as prognosis of responses to assist in damage detection and structural health monitoring.
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