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This report presents an elementary theory of unification for positive conjunctive queries. A positive conjunctive query is a formula constructed from propositional constants, equations and atoms using the conjunction $\wedge$ and the existential quantifier $\exists$. In particular, empty queries correspond to existentially quantified systems of equations -- called $\cal E$-formulas. We provide an algorithm which transforms any conjunctive query into a solved form. We prove some lattice-theoretic properties of queries. In particular, the quotient set of $\cal E$-formulas under an equivalence relation forms a complete lattice. Then we present another lattice -- a lattice of finite substitutions. We prove that the both lattices are isomorphic. Finally, we introduce the notion of application of substitutions to formulas and clarify its relationship to $\cal E$-formulas. This theory can be regarded as a basis for alternative presentation of logic programming.

Standard neural networks can approximate general nonlinear operators, represented either explicitly by a combination of mathematical operators, e.g., in an advection-diffusion-reaction partial differential equation, or simply as a black box, e.g., a system-of-systems. The first neural operator was the Deep Operator Network (DeepONet), proposed in 2019 based on rigorous approximation theory. Since then, a few other less general operators have been published, e.g., based on graph neural networks or Fourier transforms. For black box systems, training of neural operators is data-driven only but if the governing equations are known they can be incorporated into the loss function during training to develop physics-informed neural operators. Neural operators can be used as surrogates in design problems, uncertainty quantification, autonomous systems, and almost in any application requiring real-time inference. Moreover, independently pre-trained DeepONets can be used as components of a complex multi-physics system by coupling them together with relatively light training. Here, we present a review of DeepONet, the Fourier neural operator, and the graph neural operator, as well as appropriate extensions with feature expansions, and highlight their usefulness in diverse applications in computational mechanics, including porous media, fluid mechanics, and solid mechanics.

Numerically solving ordinary differential equations (ODEs) is a naturally serial process and as a result the vast majority of ODE solver software are serial. In this manuscript we developed a set of parallelized ODE solvers using extrapolation methods which exploit "parallelism within the method" so that arbitrary user ODEs can be parallelized. We describe the specific choices made in the implementation of the explicit and implicit extrapolation methods which allow for generating low overhead static schedules to then exploit with optimized multi-threaded implementations. We demonstrate that while the multi-threading gives a noticeable acceleration on both explicit and implicit problems, the explicit parallel extrapolation methods gave no significant improvement over state-of-the-art even with a multi-threading advantage against current optimized high order Runge-Kutta tableaus. However, we demonstrate that the implicit parallel extrapolation methods are able to achieve state-of-the-art performance (2x-4x) on standard multicore x86 CPUs for systems of $<200$ stiff ODEs solved at low tolerance, a typical setup for a vast majority of users of high level language equation solver suites. The resulting method is distributed as the first widely available open source software for within-method parallel acceleration targeting typical modest compute architectures.

Longitudinal biomarker data and cross-sectional outcomes are routinely collected in modern epidemiology studies, often with the goal of informing tailored early intervention decisions. For example, hormones such as estradiol and follicle-stimulating hormone may predict changes in womens' health during the midlife. Most existing methods focus on constructing predictors from mean marker trajectories. However, subject-level biomarker variability may also provide critical information about disease risks and health outcomes. In this paper, we develop a joint model that estimates subject-level means and variances of longitudinal biomarkers to predict a cross-sectional health outcome. Simulations demonstrate excellent recovery of true model parameters. The proposed method provides less biased and more efficient estimates, relative to alternative approaches that either ignore subject-level differences in variances or perform two-stage estimation where estimated marker variances are treated as observed. Analyses of women's health data reveal larger variability of E2 or larger variability of FSH were associated with higher levels of fat mass change and higher levels of lean mass change across the menopausal transition.

Claiming causal inferences in network settings necessitates careful consideration of the often complex dependency between outcomes for actors. Of particular importance are treatment spillover or outcome interference effects. We consider causal inference when the actors are connected via an underlying network structure. Our key contribution is a model for causality when the underlying network is unobserved and the actor covariates evolve stochastically over time. We develop a joint model for the relational and covariate generating process that avoids restrictive separability assumptions and deterministic network assumptions that do not hold in the majority of social network settings of interest. Our framework utilizes the highly general class of Exponential-family Random Network models (ERNM) of which Markov Random Fields (MRF) and Exponential-family Random Graph models (ERGM) are special cases. We present potential outcome based inference within a Bayesian framework, and propose a simple modification to the exchange algorithm to allow for sampling from ERNM posteriors. We present results of a simulation study demonstrating the validity of the approach. Finally, we demonstrate the value of the framework in a case-study of smoking over time in the context of adolescent friendship networks.

We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is as follows. We give a randomized algorithm, that given a colored $n$-vertex undirected graph, vertices $s$ and $t$, and an integer $k$, finds an $(s,t)$-path containing at least $k$ different colors in time $2^k n^{O(1)}$. This is the first FPT algorithm for this problem, and it generalizes the algorithm of Bj\"orklund, Husfeldt, and Taslaman [SODA 2012] on finding a path through $k$ specified vertices. It also implies the first $2^k n^{O(1)}$ time algorithm for finding an $(s,t)$-path of length at least $k$. Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an $n$-vertex undirected graph $G$, a matroid $M$ whose elements correspond to the vertices of $G$ and which is represented over a finite field of order $q$, a positive integer weight function on the vertices of $G$, two sets of vertices $S,T \subseteq V(G)$, and integers $p,k,w$, and the task is to find $p$ vertex-disjoint paths from $S$ to $T$ so that the union of the vertices of these paths contains an independent set of $M$ of cardinality $k$ and weight $w$, while minimizing the sum of the lengths of the paths. We give a $2^{p+O(k^2 \log (q+k))} n^{O(1)} w$ time randomized algorithm for this problem.

A new kind of spline geometric method approach is presented. Its main ingredient is the use of well established spline spaces forming a discrete de Rham complex to construct a primal sequence $\{X^k_h\}^n_{k=0}$, starting from splines of degree $p$, and a dual sequence $\{\tilde{X}^k_h\}_{k=0}^n$, starting from splines of degree $p-1$. By imposing homogeneous boundary conditions to the spaces of the primal sequence, the two sequences can be isomorphically mapped into one another. Within this setup, many familiar second order partial differential equations can be finally accommodated by explicitly constructing appropriate discrete versions of constitutive relations, called Hodge--star operators. Several alternatives based on both global and local projection operators between spline spaces will be proposed. The appeal of the approach with respect to similar published methods is twofold: firstly, it exhibits high order convergence. Secondly, it does not rely on the geometric realization of any (topologically) dual mesh. Several numerical examples in various space dimensions will be employed to validate the central ideas of the proposed approach and compare its features with the standard Galerkin approach in Isogeometric Analysis.

Increasing privacy concerns have given rise to Private Inference (PI). In PI, both the client's personal data and the service provider's trained model are kept confidential. State-of-the-art PI protocols combine several cryptographic primitives: Homomorphic Encryption (HE), Secret Sharing (SS), Garbled Circuits (GC), and Oblivious Transfer (OT). Today, PI remains largely arcane and too slow for practical use, despite the need and recent performance improvements. This paper addresses PI's shortcomings with a detailed characterization of a standard high-performance protocol to build foundational knowledge and intuition in the systems community. The characterization pinpoints all sources of inefficiency -- compute, communication, and storage. A notable aspect of this work is the use of inference request arrival rates rather than studying individual inferences in isolation. Prior to this work, and without considering arrival rate, it has been assumed that PI pre-computations can be handled offline and their overheads ignored. We show this is not the case. The offline costs in PI are so high that they are often incurred online, as there is insufficient downtime to hide pre-compute latency. We further propose three optimizations to address the computation (layer-parallel HE), communication (wireless slot allocation), and storage (Client-Garbler) overheads leveraging insights from our characterization. Compared to the state-of-the-art PI protocol, the optimizations provide a total PI speedup of 1.8$\times$, with the ability to sustain inference requests up to a 2.24$\times$ greater rate.

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.

Graph Neural Networks (GNNs) have proven to be useful for many different practical applications. However, many existing GNN models have implicitly assumed homophily among the nodes connected in the graph, and therefore have largely overlooked the important setting of heterophily, where most connected nodes are from different classes. In this work, we propose a novel framework called CPGNN that generalizes GNNs for graphs with either homophily or heterophily. The proposed framework incorporates an interpretable compatibility matrix for modeling the heterophily or homophily level in the graph, which can be learned in an end-to-end fashion, enabling it to go beyond the assumption of strong homophily. Theoretically, we show that replacing the compatibility matrix in our framework with the identity (which represents pure homophily) reduces to GCN. Our extensive experiments demonstrate the effectiveness of our approach in more realistic and challenging experimental settings with significantly less training data compared to previous works: CPGNN variants achieve state-of-the-art results in heterophily settings with or without contextual node features, while maintaining comparable performance in homophily settings.