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We propose and analyse boundary-preserving schemes for the strong approximations of some scalar SDEs with non-globally Lipschitz drift and diffusion coefficients whose state-space is bounded. The schemes consists of a Lamperti transform followed by a Lie--Trotter splitting. We prove $L^{p}(\Omega)$-convergence of order $1$, for every $p \geq 1$, of the schemes and exploit the Lamperti transform to confine the numerical approximations to the state-space of the considered SDE. We provide numerical experiments that confirm the theoretical results and compare the proposed Lamperti-splitting schemes to other numerical schemes for SDEs.

Quantum entanglement is a fundamental property of quantum mechanics and plays a crucial role in quantum computation and information. We study entanglement via the lens of computational complexity by considering quantum generalizations of the class NP with multiple unentangled quantum proofs, the so-called QMA(2) and its variants. The complexity of QMA(2) is a longstanding open problem, and only the trivial bounds QMA $\subseteq$ QMA(2) $\subseteq$ NEXP are known. In this work, we study the power of unentangled quantum proofs with non-negative amplitudes, a class which we denote $\text{QMA}^+(2)$. In this setting, we are able to design proof verification protocols for problems both using logarithmic size quantum proofs and having a constant probability gap in distinguishing yes from no instances. In particular, we design global protocols for small set expansion, unique games, and PCP verification. As a consequence, we obtain NP $\subseteq \text{QMA}^+_{\log}(2)$ with a constant gap. By virtue of the new constant gap, we are able to ``scale up'' this result to $\text{QMA}^+(2)$, obtaining the full characterization $\text{QMA}^+(2)$=NEXP by establishing stronger explicitness properties of the PCP for NEXP. One key novelty of these protocols is the manipulation of quantum proofs in a global and coherent way yielding constant gaps. Previous protocols (only available for general amplitudes) are either local having vanishingly small gaps or treat the quantum proofs as classical probability distributions requiring polynomially many proofs thereby not implying non-trivial bounds on QMA(2). Finally, we show that QMA(2) is equal to $\text{QMA}^+(2)$ provided the gap of the latter is a sufficiently large constant. In particular, if $\text{QMA}^+(2)$ admits gap amplification, then QMA(2)=NEXP.

The area under the ROC curve is a common measure that is often used to rank the relative performance of different binary classifiers. However, as has been also previously noted, it can be a measure that ill-captures the benefits of different classifiers when either the true class values or misclassification costs are highly unbalanced between the two classes. We introduce a third dimension to capture these costs, and lift the ROC curve to a ROC surface in a natural way. We study both this surface and introduce the VOROS, the volume over this ROC surface, as a 3D generalization of the 2D area under the ROC curve. For problems where there are only bounds on the expected costs or class imbalances, we restrict consideration to the volume of the appropriate subregion of the ROC surface. We show how the VOROS can better capture the costs of different classifiers on both a classical and a modern example dataset.

Fully decentralized learning is gaining momentum for training AI models at the Internet's edge, addressing infrastructure challenges and privacy concerns. In a decentralized machine learning system, data is distributed across multiple nodes, with each node training a local model based on its respective dataset. The local models are then shared and combined to form a global model capable of making accurate predictions on new data. Our exploration focuses on how different types of network structures influence the spreading of knowledge - the process by which nodes incorporate insights gained from learning patterns in data available on other nodes across the network. Specifically, this study investigates the intricate interplay between network structure and learning performance using three network topologies and six data distribution methods. These methods consider different vertex properties, including degree centrality, betweenness centrality, and clustering coefficient, along with whether nodes exhibit high or low values of these metrics. Our findings underscore the significance of global centrality metrics (degree, betweenness) in correlating with learning performance, while local clustering proves less predictive. We highlight the challenges in transferring knowledge from peripheral to central nodes, attributed to a dilution effect during model aggregation. Additionally, we observe that central nodes exert a pull effect, facilitating the spread of knowledge. In examining degree distribution, hubs in Barabasi-Albert networks positively impact learning for central nodes but exacerbate dilution when knowledge originates from peripheral nodes. Finally, we demonstrate the formidable challenge of knowledge circulation outside of segregated communities.

The multiobjective evolutionary optimization algorithm (MOEA) is a powerful approach for tackling multiobjective optimization problems (MOPs), which can find a finite set of approximate Pareto solutions in a single run. However, under mild regularity conditions, the Pareto optimal set of a continuous MOP could be a low dimensional continuous manifold that contains infinite solutions. In addition, structure constraints on the whole optimal solution set, which characterize the patterns shared among all solutions, could be required in many real-life applications. It is very challenging for existing finite population based MOEAs to handle these structure constraints properly. In this work, we propose the first model-based algorithmic framework to learn the whole solution set with structure constraints for multiobjective optimization. In our approach, the Pareto optimality can be traded off with a preferred structure among the whole solution set, which could be crucial for many real-world problems. We also develop an efficient evolutionary learning method to train the set model with structure constraints. Experimental studies on benchmark test suites and real-world application problems demonstrate the promising performance of our proposed framework.

We propose a two-scale neural network method for solving partial differential equations (PDEs) with small parameters using physics-informed neural networks (PINNs). We directly incorporate the small parameters into the architecture of neural networks. The proposed method enables solving PDEs with small parameters in a simple fashion, without adding Fourier features or other computationally taxing searches of truncation parameters. Various numerical examples demonstrate reasonable accuracy in capturing features of large derivatives in the solutions caused by small parameters.

We introduce generative models for accelerating simulations of complex systems through learning and evolving their effective dynamics. In the proposed Generative Learning of Effective Dynamics (G-LED), instances of high dimensional data are down sampled to a lower dimensional manifold that is evolved through an auto-regressive attention mechanism. In turn, Bayesian diffusion models, that map this low-dimensional manifold onto its corresponding high-dimensional space, capture the statistics of the system dynamics. We demonstrate the capabilities and drawbacks of G-LED in simulations of several benchmark systems, including the Kuramoto-Sivashinsky (KS) equation, two-dimensional high Reynolds number flow over a backward-facing step, and simulations of three-dimensional turbulent channel flow. The results demonstrate that generative learning offers new frontiers for the accurate forecasting of the statistical properties of complex systems at a reduced computational cost.

Data assimilation (DA) methods use priors arising from differential equations to robustly interpolate and extrapolate data. Popular techniques such as ensemble methods that handle high-dimensional, nonlinear PDE priors focus mostly on state estimation, however can have difficulty learning the parameters accurately. On the other hand, machine learning based approaches can naturally learn the state and parameters, but their applicability can be limited, or produce uncertainties that are hard to interpret. Inspired by the Integrated Nested Laplace Approximation (INLA) method in spatial statistics, we propose an alternative approach to DA based on iteratively linearising the dynamical model. This produces a Gaussian Markov random field at each iteration, enabling one to use INLA to infer the state and parameters. Our approach can be used for arbitrary nonlinear systems, while retaining interpretability, and is furthermore demonstrated to outperform existing methods on the DA task. By providing a more nuanced approach to handling nonlinear PDE priors, our methodology offers improved accuracy and robustness in predictions, especially where data sparsity is prevalent.

Mesh degeneration is a bottleneck for fluid-structure interaction (FSI) simulations and for shape optimization via the method of mappings. In both cases, an appropriate mesh motion technique is required. The choice is typically based on heuristics, e.g., the solution operators of partial differential equations (PDE), such as the Laplace or biharmonic equation. Especially the latter, which shows good numerical performance for large displacements, is expensive. Moreover, from a continuous perspective, choosing the mesh motion technique is to a certain extent arbitrary and has no influence on the physically relevant quantities. Therefore, we consider approaches inspired by machine learning. We present a hybrid PDE-NN approach, where the neural network (NN) serves as parameterization of a coefficient in a second order nonlinear PDE. We ensure existence of solutions for the nonlinear PDE by the choice of the neural network architecture. Moreover, we present an approach where a neural network corrects the harmonic extension such that the boundary displacement is not changed. In order to avoid technical difficulties in coupling finite element and machine learning software, we work with a splitting of the monolithic FSI system into three smaller subsystems. This allows to solve the mesh motion equation in a separate step. We assess the quality of the learned mesh motion technique by applying it to a FSI benchmark problem. In addition, we discuss generalizability and computational cost of the learned mesh motion operators.

Interior point methods are widely used for different types of mathematical optimization problems. Many implementations of interior point methods in use today rely on direct linear solvers to solve systems of equations in each iteration. The need to solve ever larger optimization problems more efficiently and the rise of hardware accelerators for general purpose computing has led to a large interest in using iterative linear solvers instead, with the major issue being inevitable ill-conditioning of the linear systems arising as the optimization progresses. We investigate the use of Krylov solvers for interior point methods in solving optimization problems from radiation therapy and support vector machines. We implement a prototype interior point method using a so called doubly augmented formulation of the Karush-Kuhn-Tucker linear system of equations, originally proposed by Forsgren and Gill, and evaluate its performance on real optimization problems from radiation therapy and support vector machines. Crucially, our implementation uses a preconditioned conjugate gradient method with Jacobi preconditioning internally. Our measurements of the conditioning of the linear systems indicate that the Jacobi preconditioner improves the conditioning of the systems to a degree that they can be solved iteratively, but there is room for further improvement in that regard. Furthermore, profiling of our prototype code shows that it is suitable for GPU acceleration, which may further improve its performance in practice. Overall, our results indicate that our method can find solutions of acceptable accuracy in reasonable time, even with a simple Jacobi preconditioner.