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Process curves are multivariate finite time series data coming from manufacturing processes. This paper studies machine learning that detect drifts in process curve datasets. A theoretic framework to synthetically generate process curves in a controlled way is introduced in order to benchmark machine learning algorithms for process drift detection. An evaluation score, called the temporal area under the curve, is introduced, which allows to quantify how well machine learning models unveil curves belonging to drift segments. Finally, a benchmark study comparing popular machine learning approaches on synthetic data generated with the introduced framework is presented that shows that existing algorithms often struggle with datasets containing multiple drift segments.

Sparse parametric models are of great interest in statistical learning and are often analyzed by means of regularized estimators. Pathwise methods allow to efficiently compute the full solution path for penalized estimators, for any possible value of the penalization parameter $\lambda$. In this paper we deal with the pathwise optimization for bridge-type problems; i.e. we are interested in the minimization of a loss function, such as negative log-likelihood or residual sum of squares, plus the sum of $\ell^q$ norms with $q\in(0,1]$ involving adpative coefficients. For some loss functions this regularization achieves asymptotically the oracle properties (such as the selection consistency). Nevertheless, since the objective function involves nonconvex and nondifferentiable terms, the minimization problem is computationally challenging. The aim of this paper is to apply some general algorithms, arising from nonconvex optimization theory, to compute efficiently the path solutions for the adaptive bridge estimator with multiple penalties. In particular, we take into account two different approaches: accelerated proximal gradient descent and blockwise alternating optimization. The convergence and the path consistency of these algorithms are discussed. In order to assess our methods, we apply these algorithms to the penalized estimation of diffusion processes observed at discrete times. This latter represents a recent research topic in the field of statistics for time-dependent data.

We propose deep learning methods for classical Monge's optimal mass transportation problems, where where the distribution constraint is treated as penalty terms defined by the maximum mean discrepancy in the theory of Hilbert space embeddings of probability measures. We prove that the transport maps given by the proposed methods converge to optimal transport maps in the problem with $L^2$ cost. Several numerical experiments validate our methods. In particular, we show that our methods are applicable to large-scale Monge problems.

We propose a framework to learn the time-dependent Hartree-Fock (TDHF) inter-electronic potential of a molecule from its electron density dynamics. Though the entire TDHF Hamiltonian, including the inter-electronic potential, can be computed from first principles, we use this problem as a testbed to develop strategies that can be applied to learn a priori unknown terms that arise in other methods/approaches to quantum dynamics, e.g., emerging problems such as learning exchange-correlation potentials for time-dependent density functional theory. We develop, train, and test three models of the TDHF inter-electronic potential, each parameterized by a four-index tensor of size up to $60 \times 60 \times 60 \times 60$. Two of the models preserve Hermitian symmetry, while one model preserves an eight-fold permutation symmetry that implies Hermitian symmetry. Across seven different molecular systems, we find that accounting for the deeper eight-fold symmetry leads to the best-performing model across three metrics: training efficiency, test set predictive power, and direct comparison of true and learned inter-electronic potentials. All three models, when trained on ensembles of field-free trajectories, generate accurate electron dynamics predictions even in a field-on regime that lies outside the training set. To enable our models to scale to large molecular systems, we derive expressions for Jacobian-vector products that enable iterative, matrix-free training.

Many partial differential equations (PDEs) such as Navier--Stokes equations in fluid mechanics, inelastic deformation in solids, and transient parabolic and hyperbolic equations do not have an exact, primal variational structure. Recently, a variational principle based on the dual (Lagrange multiplier) field was proposed. The essential idea in this approach is to treat the given PDE as constraints, and to invoke an arbitrarily chosen auxiliary potential with strong convexity properties to be optimized. This leads to requiring a convex dual functional to be minimized subject to Dirichlet boundary conditions on dual variables, with the guarantee that even PDEs that do not possess a variational structure in primal form can be solved via a variational principle. The vanishing of the first variation of the dual functional is, up to Dirichlet boundary conditions on dual fields, the weak form of the primal PDE problem with the dual-to-primal change of variables incorporated. We derive the dual weak form for the linear, one-dimensional, transient convection-diffusion equation. A Galerkin discretization is used to obtain the discrete equations, with the trial and test functions chosen as linear combination of either RePU activation functions (shallow neural network) or B-spline basis functions; the corresponding stiffness matrix is symmetric. For transient problems, a space-time Galerkin implementation is used with tensor-product B-splines as approximating functions. Numerical results are presented for the steady-state and transient convection-diffusion equation, and transient heat conduction. The proposed method delivers sound accuracy for ODEs and PDEs and rates of convergence are established in the $L^2$ norm and $H^1$ seminorm for the steady-state convection-diffusion problem.

This paper presents a unifying framework for Trefftz-like methods, which allows the analysis and construction of discretization methods based on the decomposition into, and coupling of, local and global problems. We apply the framework to provide a comprehensive error analysis for the Embedded Trefftz discontinuous Galerkin method, for a wide range of second-order scalar elliptic partial differential equations and a scalar reaction-advection problem. We also analyze quasi-Trefftz methods with our framework and build bridges to other methods that are similar in virtue.

We consider statistical inference for a class of mixed-effects models with system noise described by a non-Gaussian integrated Ornstein-Uhlenbeck process. Under the asymptotics where the number of individuals goes to infinity with possibly unbalanced sampling frequency across individuals, we prove some theoretical properties of the Gaussian quasi-likelihood function, followed by the asymptotic normality and the tail-probability estimate of the associated estimator. In addition to the joint inference, we propose and investigate the three-stage inference strategy, revealing that they are first-order equivalent while quantitatively different in the second-order terms. Numerical experiments are given to illustrate the theoretical results.

Control structure design is an important but tedious step in P&ID development. Generative artificial intelligence (AI) promises to reduce P&ID development time by supporting engineers. Previous research on generative AI in chemical process design mainly represented processes by sequences. However, graphs offer a promising alternative because of their permutation invariance. We propose the Graph-to-SFILES model, a generative AI method to predict control structures from flowsheet topologies. The Graph-to-SFILES model takes the flowsheet topology as a graph input and returns a control-extended flowsheet as a sequence in the SFILES 2.0 notation. We compare four different graph encoder architectures, one of them being a graph neural network (GNN) proposed in this work. The Graph-to-SFILES model achieves a top-5 accuracy of 73.2% when trained on 10,000 flowsheet topologies. In addition, the proposed GNN performs best among the encoder architectures. Compared to a purely sequence-based approach, the Graph-to-SFILES model improves the top-5 accuracy for a relatively small training dataset of 1,000 flowsheets from 0.9% to 28.4%. However, the sequence-based approach performs better on a large-scale dataset of 100,000 flowsheets. These results highlight the potential of graph-based AI models to accelerate P&ID development in small-data regimes but their effectiveness on industry relevant case studies still needs to be investigated.

Combinatorial problems such as combinatorial optimization and constraint satisfaction problems arise in decision-making across various fields of science and technology. In real-world applications, when multiple optimal or constraint-satisfying solutions exist, enumerating all these solutions -- rather than finding just one -- is often desirable, as it provides flexibility in decision-making. However, combinatorial problems and their enumeration versions pose significant computational challenges due to combinatorial explosion. To address these challenges, we propose enumeration algorithms for combinatorial optimization and constraint satisfaction problems using Ising machines. Ising machines are specialized devices designed to efficiently solve combinatorial problems. Typically, they sample low-cost solutions in a stochastic manner. Our enumeration algorithms repeatedly sample solutions to collect all desirable solutions. The crux of the proposed algorithms is their stopping criteria for sampling, which are derived based on probability theory. In particular, the proposed algorithms have theoretical guarantees that the failure probability of enumeration is bounded above by a user-specified value, provided that lower-cost solutions are sampled more frequently and equal-cost solutions are sampled with equal probability. Many physics-based Ising machines are expected to (approximately) satisfy these conditions. As a demonstration, we applied our algorithm using simulated annealing to maximum clique enumeration on random graphs. We found that our algorithm enumerates all maximum cliques in large dense graphs faster than a conventional branch-and-bound algorithm specially designed for maximum clique enumeration. This demonstrates the promising potential of our proposed approach.

We present an efficient algorithm for the application of sequences of planar rotations to a matrix. Applying such sequences efficiently is important in many numerical linear algebra algorithms for eigenvalues. Our algorithm is novel in three main ways. First, we introduce a new kernel that is optimized for register reuse in a novel way. Second, we introduce a blocking and packing scheme that improves the cache efficiency of the algorithm. Finally, we thoroughly analyze the memory operations of the algorithm which leads to important theoretical insights and makes it easier to select good parameters. Numerical experiments show that our algorithm outperforms the state-of-the-art and achieves a flop rate close to the theoretical peak on modern hardware.