arXiv subjects

Jonathan Gorard

Publications and source records attributed to Jonathan Gorard.

At least 19 recordsLinked to original sources

arXiv2026

BEACONS: Bounded-Error, Algebraically-Composable Neural Solvers for Partial Differential Equations

The traditional limitations of neural networks in reliably generalizing beyond the convex hulls of their training data present a significant problem for computational physics, in which one often wishes to solve PDEs in regimes far beyond anything which can be experimentally or analytically validated. In this paper, we show how it is possible to circumvent these limitations by constructing formally-verified neural network solvers for PDEs, with rigorous convergence, stability, and conservation properties, whose correctness can therefore be guaranteed even in extrapolatory regimes. By using the method of characteristics to predict the analytical properties of PDE solutions a priori (even in regions arbitrarily far from the training domain), we show how it is possible to construct rigorous extrapolatory bounds on the worst-case L^inf errors of shallow neural network approximations. Then, by decomposing PDE solutions into compositions of simpler functions, we show how it is possible to compose these shallow neural networks together to form deep architectures, based on ideas from compositional deep learning, in which the large L^inf errors in the approximations have been suppressed. The resulting framework, called BEACONS (Bounded-Error, Algebraically-COmposable Neural Solvers), comprises both an automatic code-generator for the neural solvers themselves, as well as a bespoke automated theorem-proving system for producing machine-checkable certificates of correctness. We apply the framework to a variety of linear and non-linear PDEs, including the linear advection and inviscid Burgers' equations, as well as the full compressible Euler equations, in both 1D and 2D, and illustrate how BEACONS architectures are able to extrapolate solutions far beyond the training data in a reliable and bounded way. Various advantages of the approach over the classical PINN approach are discussed.

cs.LG↗

arXiv2025

Beyond GRMHD: A Robust Numerical Scheme for Extended, Non-Ideal General Relativistic Multifluid Simulations

The equations of general relativistic magnetohydrodynamics (GRMHD) have become the standard mathematical framework for modeling high-energy plasmas in curved spacetimes. However, the fragility of the primitive variable reconstruction operation in GRMHD, as well as the difficulties in maintaining strong hyperbolicity of the equations, sharply limit the applicability of the GRMHD model in scenarios involving large Lorentz factors and high magnetizations, such as around neutron stars. Non-ideal effects, such as electron inertia and Hall terms, are also neglected, and the absence of an explicitly evolved electric field precludes the self-consistent modeling of the strong poloidal fields found around spinning black holes, which are known to be crucial for jet formation. Here, we present a general relativistic multifluid model which strictly generalizes the GRMHD equations, consisting of an arbitrary number of relativistic fluid species interacting with a shared electromagnetic field via an explicit coupling of their source terms, thus allowing for the incorporation of non-ideal effects. We sketch how our model may be derived from general relativistic kinetics (via moments of the relativistic Boltzmann-Vlasov equation), as well as how GRMHD may be recovered in the single-fluid limit as the mobility of charge carriers goes to infinity. We present a numerical scheme for solving the general relativistic multifluid equations, and validate it against the analogous scheme for the GRMHD equations. Since the primitive variable reconstruction operation for our multifluid model is purely hydrodynamic, and therefore independent of the magnetic field, the resulting solver is highly robust, and able to simulate significantly larger Lorentz factors and higher magnetizations (across both black hole and neutron star spacetimes) than GRMHD without loss of either accuracy or stability.

gr-qc↗

arXiv2025

Hydrodynamic and Electromagnetic Discrepancies between Neutron Star and Black Hole Spacetimes

The exterior spacetime geometry surrounding an uncharged, spinning black hole in general relativity depends only upon its mass and spin. However, the exterior geometry surrounding any other rotating compact object, for example a neutron star, will generally depend upon higher moments in its multipole expansion, which will in turn be dependent upon the object's equation of state. Using general relativistic hydrodynamics and electrodynamics simulations, we illustrate that the presence or absence of these higher moments (assuming a physically realistic neutron star equation of state) has a significant qualitative effect near the surface of the compact object on the dynamics of unmagnetized accretion, and a smaller quantitative effect on the electromagnetic field configuration of its magnetosphere. In some places, the discrepancies in energy-momentum density are found to reach or exceed 50%, with electric field strength discrepancies in excess of 10%. We argue that many of these differences are likely to be amplified by the inclusion of more sophisticated plasma physics models, and are therefore likely to be relevant for the dynamics of gravitational collapse, and potentially also for particle acceleration and jet launching. These discrepancies suggest important limitations regarding the use of the Kerr metric when performing numerical simulations around neutron stars.

gr-qc↗

arXiv2025

Improved Dimensionality Reduction for Inverse Problems in Nuclear Fusion and High-Energy Astrophysics

Many inverse problems in nuclear fusion and high-energy astrophysics research, such as the optimization of tokamak reactor geometries or the inference of black hole parameters from interferometric images, necessitate high-dimensional parameter scans and large ensembles of simulations to be performed. Such inverse problems typically involve large uncertainties, both in the measurement parameters being inverted and in the underlying physics models themselves. Monte Carlo sampling, when combined with modern non-linear dimensionality reduction techniques such as autoencoders and manifold learning, can be used to reduce the size of the parameter spaces considerably. However, there is no guarantee that the resulting combinations of parameters will be physically valid, or even mathematically consistent. In this position paper, we advocate adopting a hybrid approach that leverages our recent advances in the development of formal verification methods for numerical algorithms, with the goal of constructing parameter space restrictions with provable mathematical and physical correctness properties, whilst nevertheless respecting both experimental uncertainties and uncertainties in the underlying physical processes.

cs.LG↗

arXiv2025

Shock with Confidence: Formal Proofs of Correctness for Hyperbolic Partial Differential Equation Solvers

First-order systems of hyperbolic partial differential equations (PDEs) occur ubiquitously throughout computational physics, commonly used in simulations of fluid turbulence, shock waves, electromagnetic interactions, and even general relativistic phenomena. Such equations are often challenging to solve numerically in the non-linear case, due to their tendency to form discontinuities even for smooth initial data, which can cause numerical algorithms to become unstable, violate conservation laws, or converge to physically incorrect solutions. In this paper, we introduce a new formal verification pipeline for such algorithms in Racket, which allows a user to construct a bespoke hyperbolic PDE solver for a specified equation system, generate low-level C code which verifiably implements that solver, and then produce formal proofs of various mathematical and physical correctness properties of the resulting implementation, including L^2 stability, flux conservation, and physical validity. We outline how these correctness proofs are generated, using a custom-built theorem-proving and automatic differentiation framework that fully respects the algebraic structure of floating-point arithmetic, and show how the resulting C code may either be used to run standalone simulations, or integrated into a larger computational multiphysics framework such as Gkeyll.

cs.LO↗

arXiv2024

A Tetrad-First Approach to Robust Numerical Algorithms in General Relativity

General relativistic Riemann solvers are typically complex, fragile and unwieldy, at least in comparison to their special relativistic counterparts. In this paper, we present a new high-resolution shock-capturing algorithm on curved spacetimes that employs a local coordinate transformation at each inter-cell boundary, transforming all primitive and conservative variables into a locally flat spacetime coordinate basis (i.e., the tetrad basis), generalizing previous approaches developed for relativistic hydrodynamics. This algorithm enables one to employ a purely special relativistic Riemann solver, combined with an appropriate post-hoc flux correction step, irrespective of the geometry of the underlying Lorentzian manifold. We perform a systematic validation of the algorithm using the Gkeyll simulation framework for both general relativistic electromagnetism and general relativistic hydrodynamics, highlighting the algorithm's superior convergence and stability properties in each case when compared against standard analytical solutions for black hole magnetosphere and ultra-relativistic black hole accretion problems. However, as an illustration of the generality and practicality of the algorithm, we also apply it to more astrophysically realistic magnetosphere and fluid accretion problems in the limit of high black hole spin, for which standard general relativistic Riemann solvers are often too unstable to produce useful solutions.

gr-qc↗

arXiv2024

Applied Category Theory in the Wolfram Language using Categorica I: Diagrams, Functors and Fibrations

This article serves as a preliminary introduction to the design of a new, open-source applied and computational category theory framework, named Categorica, built on top of the Wolfram Language. Categorica allows one to configure and manipulate abstract quivers, categories, groupoids, diagrams, functors and natural transformations, and to perform a vast array of automated abstract algebraic computations using (arbitrary combinations of) the above structures; to manipulate and abstractly reason about arbitrary universal properties, including products, coproducts, pullbacks, pushouts, limits and colimits; and to manipulate, visualize and compute with strict (symmetric) monoidal categories, including full support for automated string diagram rewriting and diagrammatic theorem-proving. In so doing, Categorica combines the capabilities of an abstract computer algebra framework (thus allowing one to compute directly with epimorphisms, monomorphisms, retractions, sections, spans, cospans, fibrations, etc.) with those of a powerful automated theorem-proving system (thus allowing one to convert universal properties and other abstract constructions into (higher-order) equational logic statements that can be reasoned about and proved using standard automated theorem-proving methods, as well as to prove category-theoretic statements directly using purely diagrammatic methods). In this first of two articles introducing the design of the framework, we shall focus principally upon its handling of quivers, categories, diagrams, groupoids, functors and natural transformations, including demonstrations of both its algebraic manipulation and theorem-proving capabilities in each case.

math.CT↗

arXiv2024

General Relativistic Hydrodynamics in Discrete Spacetime: Perfect Fluid Accretion onto Static and Spinning Black Holes

We study the problem of a spherically-symmetric distribution of a perfect relativistic fluid accreting onto a (potentially spinning) black hole within a fully discrete spacetime setting. This problem has previously been studied extensively in the context of continuum spacetimes, beginning with the purely analytic work of Bondi in the spherically-symmetric Newtonian case, Michel in the spherically-symmetric general relativistic case, and Petrich, Shapiro and Teukolsky in the axially-symmetric general relativistic case relevant for spinning black holes. However, the purpose of the present work is to determine the effect of discretization of the underlying spacetime upon the mass/energy and momentum accretion rates, the overall morphology and characteristics of the accretion flow, and the drag force exerted on the black hole in the case of non-zero spin. In order to achieve this, we first develop a novel formulation of the equations of general relativistic hydrodynamics that is more directly amenable to rigorous analysis within a discrete spacetime setting, and we then proceed to implement this formulation into the Gravitas computational general relativity framework. Through a combination of mathematical analysis and explicit numerical simulation in Gravitas, we discover that the mass/energy and momentum accretion rates both decrease monotonically as functions of the underlying spacetime discretization scale, with this effect becoming more pronounced for higher values of the black hole spin parameter, higher fluid temperatures, and stiffer equation of state parameters. We also find that the exerted drag force is highly sensitive to the value of the underlying discretization scale in the case of spinning black hole spacetimes, with certain instabilities becoming significantly more pronounced at certain critical values of the discretization parameter.

gr-qc↗

arXiv2024

Computational General Relativity in the Wolfram Language using Gravitas II: ADM Formalism and Numerical Relativity

This is the second in a series of two articles introducing the Gravitas computational general relativity framework, in which we now focus upon the design and capabilities of Gravitas's numerical subsystem, including its ability to perform general 3+1 decompositions of spacetime via the ADM formalism, its support for the definition and construction of arbitrary Cauchy surfaces as initial data, its support for the definition and enforcement of arbitrary gauge and coordinate conditions, its various algorithms for ensuring the satisfaction of the ADM Hamiltonian and momentum constraints, and its unique adaptive refinement algorithms based on hypergraph rewriting via Wolfram model evolution. Particular attention is paid to the seamless integration between Gravitas's symbolic and numerical subsystems, its ability to configure, run, analyze and visualize complex numerical relativity simulations and their outputs within a single notebook environment, and its capabilities for handling generic curvilinear coordinate systems and spacetimes with general (and often highly non-trivial) topologies using its specialized and highly efficient hypergraph-based numerical algorithms. We also provide illustrations of Gravitas's functionality for the visualization of hypergraph geometries and spacetime embedding diagrams, the ability for Gravitas's symbolic and numerical subsystems to be used in concert for the extraction of gravitational wave signals and other crucial simulation data, and Gravitas's in-built library of standard initial data, matter distributions and gauge conditions. We conclude by demonstrating how the numerical subsystem can be used to set up, run, visualize and analyze a standard yet nevertheless reasonably challenging numerical relativity test case: a binary black hole collision and merger within a vacuum spacetime (including the extraction of its outgoing gravitational wave profile).

gr-qc↗

arXiv2023

Computational General Relativity in the Wolfram Language using Gravitas I: Symbolic and Analytic Computation

We introduce a new, open-source computational general relativity framework for the Wolfram Language called Gravitas, which boasts a number of novel and distinctive features as compared to the many pre-existing computational and numerical relativity frameworks currently available within the open-source community. These include, but are not limited to: seamless integration of its powerful symbolic and numerical subsystems, and, by extension, seamless transition between analytic/continuous representations and numerical/discrete representations of arbitrary spacetime geometries; highly modular, general and extensible representations of spacetime geometries, spacetime topologies, gauge conditions, coordinate systems, matter fields, evolution equations and initial data; ability to set up and run complex numerical relativity simulations, and to perform 2D and 3D visualizations, symbolic computations and numerical analysis (including the extraction of gravitational wave signals) on the resulting data, all from within a single notebook environment; and a totally-unstructured adaptive refinement scheme based on hypergraph rewriting, allowing for exceedingly efficient discretization and numerical evolution of Cauchy initial data for a wide range of challenging computational problems involving strong relativistic field dynamics. In this first in a series of two articles covering the framework, we focus on the design and capabilities of Gravitas's symbolic subsystem, including its general and flexible handling of arbitrary geometries parametrized by arbitrary curvilinear coordinate systems (along with an in-built library of standard metrics and coordinate conditions), as well as its various high-level tensor calculus and differential geometry features. We proceed to show how this subsystem can be used to solve the Einstein field equations both analytically and numerically.

gr-qc↗

arXiv2023

Non-Vacuum Solutions, Gravitational Collapse and Discrete Singularity Theorems in Wolfram Model Systems

The celebrated geodesic congruence equation of Raychaudhuri, together with the resulting singularity theorems of Penrose and Hawking that it enabled, yield a highly general set of conditions under which a spacetime (or, more generically, a pseudo-Riemannian manifold) is expected to become geodesically incomplete. It is a non-trivial question to ask whether (and to what extent) there exist equivalently general conditions under which one expects discrete spacetimes to become geodesically incomplete, and how these conditions might differ from those in the continuum. This article builds upon previous work, in which the conformal and covariant Z4 (CCZ4) formulation of the Cauchy problem for the Einstein field equations, with constraint-violation damping, was defined in terms of Wolfram model evolution over discrete (spatial) hypergraphs for the case of vacuum spacetimes, and proceeds to consider a minimal extension to the non-vacuum case by introducing a massive scalar field distribution, defined in either spherical or axial symmetry. Under appropriate assumptions, this scalar field distribution admits a physical interpretation as a collapsing (and, in the axially-symmetric case, uniformly rotating) dust, and we are able to show, through a combination of rigorous mathematical analysis and explicit numerical simulation, that the resulting discrete spacetimes converge asymptotically to either non-rotating Schwarzschild black hole solutions or maximally-rotating (extremal) Kerr black hole solutions, respectively. Although the assumptions used in obtaining these preliminary results are very strong, they nevertheless offer hope that a more general, perhaps ultimately ``Penrose-like'', singularity theorem may be provable in the discrete spacetime case too.

gr-qc↗

arXiv2023

Axiomatic Quantum Field Theory in Discrete Spacetime via Multiway Causal Structure: The Case of Entanglement Entropies

The causal set and Wolfram model approaches to discrete quantum gravity both permit the formulation of a manifestly covariant notion of entanglement entropy for quantum fields. In the causal set case, this is given by a construction (due to Sorkin and Johnston) of a 2-point correlation function for a Gaussian scalar field from causal set Feynman propagators and Pauli-Jordan functions, from which an eigendecomposition, and hence an entanglement entropy, can be computed. In the Wolfram model case, it is given instead in terms of the Fubini-Study metric on branchial graphs, whose tensor product structure is inherited functorially from that of finite-dimensional Hilbert spaces. In both cases, the entanglement entropies in question are most naturally defined over an extended spacetime region (hence the manifest covariance), in contrast to the generically non-covariant definitions over single spacelike hypersurfaces common to most continuum quantum field theories. In this article, we show how an axiomatic field theory for a free, massless scalar field (obeying the appropriate bosonic commutation relations) may be rigorously constructed over multiway causal graphs: a combinatorial structure sufficiently general as to encompass both causal sets and Wolfram model evolutions as special cases. We proceed to show numerically that the entanglement entropies computed using both the Sorkin-Johnston approach and the branchial graph approach are monotonically related for a large class of Wolfram model evolution rules. We also prove a special case of this monotonic relationship using a recent geometrical entanglement monotone proposed by Cocchiarella et al.

gr-qc↗

arXiv2022

A Functorial Perspective on (Multi)computational Irreducibility

This article aims to provide a novel formalization of the concept of computational irreducibility in terms of the exactness of functorial correspondence between a category of data structures and elementary computations and a corresponding category of (1-dimensional) cobordisms. We proceed to demonstrate that, by equipping both categories with a symmetric monoidal structure and considering the case of higher-dimensional cobordism categories, we obtain a natural extension of this formalism that serves also to encompass non-deterministic or ``multiway'' computations, in which one quantifies not only the irreducibility in the behavior of a single (deterministic) computation path, but in the branching and merging behavior of an entire ``multiway system'' of such paths too. We finally outline how, in the most general case, the resulting symmetric monoidal functor may be considered to be adjoint to the functor characterizing the Atiyah-Segal axiomatization of a functorial quantum field theory. Thus, we conclude by arguing that the irreducibility of (multi)computations may be thought of as being dual to the locality of time evolution in functorial approaches to quantum mechanics and quantum field theory. In the process, we propose an extension of the methods of standard (monoidal) category theory, in which morphisms are effectively equipped with intrinsic computational complexity data, together with an algebra for how those complexities compose (both in sequence and in parallel, subject to the monoidal structure).

cs.CC↗

arXiv2021

Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories

We investigate algebraic and compositional properties of abstract multiway rewriting systems, which are archetypical structures underlying the formalism of the Wolfram model. We demonstrate the existence of higher homotopies in this class of rewriting systems, where homotopical maps are induced by the inclusion of appropriate rewriting rules taken from an abstract rulial space of all possible such rules. Furthermore, we show that a multiway rewriting system with homotopies up to order $n$ may naturally be formalized as an $n$-fold category, such that (upon inclusion of appropriate inverse morphisms via invertible rewriting relations) the infinite limit of this structure yields an ${\infty}$-groupoid. Via Grothendieck's homotopy hypothesis, this ${\infty}$-groupoid thus inherits the structure of a formal homotopy space. We conclude with some comments on how this computational framework of homotopical multiway systems may potentially be used for making formal connections to homotopy spaces upon which models relevant to physics may be instantiated.

math.CT↗

arXiv2021

Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types

How do spaces emerge from pregeometric discrete building blocks governed by computational rules? To address this, we investigate non-deterministic rewriting systems (multiway systems) of the Wolfram model. We express these rewriting systems as homotopy types. Using this new formulation, we outline how spatial structures can be functorially inherited from pregeometric type-theoretic constructions. We show how higher homotopy types are constructed from rewriting rules. These correspond to morphisms of an $n$-fold category. Subsequently, the $n \to \infty$ limit of the Wolfram model rulial multiway system is identified as an $\infty$-groupoid, with the latter being relevant given Grothendieck's homotopy hypothesis. We then go on to show how this construction extends to the classifying space of rulial multiway systems, which forms a multiverse of multiway systems and carries the formal structure of an ${\left(\infty, 1\right)}$-topos. This correspondence to higher categorical structures offers a new way to understand how spaces relevant to physics may arise from pregeometric combinatorial models. A key issue we have addressed here is to relate abstract non-deterministic rewriting systems to higher homotopy spaces. A consequence of constructing spaces and geometry synthetically is that it eliminates ad hoc assumptions about geometric attributes of a model such as an a priori background or pre-assigned geometric data. Instead, geometry is inherited functorially by higher structures. This is relevant for formally justifying different choices of underlying spacetime discretization adopted by models of quantum gravity. We conclude with comments on how our framework of higher category-theoretic combinatorial constructions, corroborates with other approaches investigating higher categorical structures relevant to the foundations of physics.

math.CT↗

arXiv2021

Algorithmic Causal Sets and the Wolfram Model

The formal relationship between two differing approaches to the description of spacetime as an intrinsically discrete mathematical structure, namely causal set theory and the Wolfram model, is studied, and it is demonstrated that the hypergraph rewriting approach of the Wolfram model can effectively be interpreted as providing an underlying algorithmic dynamics for causal set evolution. We show how causal invariance of the hypergraph rewriting system can be used to infer conformal invariance of the induced causal partial order, in a manner that is provably compatible with the measure-theoretic arguments of Bombelli, Henson and Sorkin. We then illustrate how many of the local dimension estimation algorithms developed in the context of the Wolfram model may be reformulated as generalizations of the midpoint scaling estimator on causal sets, and are compatible with the generalized Myrheim-Meyer estimators, as well as exploring how the presence of the underlying hypergraph structure yields a significantly more robust technique for estimating spacelike distances when compared against several standard distance and predistance estimator functions in causal set theory. We finally demonstrate how the Benincasa-Dowker action on causal sets can be recovered as a special case of the discrete Einstein-Hilbert action over Wolfram model systems (with ergodicity assumptions in the hypergraph replaced by Poisson distribution assumptions in the causal set), and also how both classical and quantum sequential growth dynamics can be recovered as special cases of Wolfram model multiway evolution with an appropriate choice of discrete measure.

gr-qc↗

arXiv2021

Hypergraph Discretization of the Cauchy Problem in General Relativity via Wolfram Model Evolution

Although the traditional form of the Einstein field equations is intrinsically four-dimensional, the field of numerical general relativity focuses on the reformulation of these equations as a 3 + 1-dimensional Cauchy problem, in which Cauchy initial data is specified on a three-dimensional spatial hypersurface, and then evolved forwards in time. The Wolfram model offers an inherently discrete formulation of the Einstein field equations as an a priori Cauchy problem, in which Cauchy initial data is specified on a single spatial hypergraph, and then evolved by means of hypergraph substitution rules, with the resulting causal network corresponding to the conformal structure of spacetime. This article introduces a new numerical general relativity code based upon the conformal and covariant Z4 (CCZ4) formulation with constraint-violation damping, with the option to reduce to the standard BSSN formalism if desired, with Cauchy data defined over hypergraphs; the code incorporates an unstructured generalization of the adaptive mesh refinement technique proposed by Berger and Colella, in which the topology of the hypergraph is refined or coarsened based upon local conformal curvature terms. We validate this code numerically against a variety of standard spacetimes, including Schwarzschild black holes, Kerr black holes, maximally extended Schwarzschild black holes, and binary black hole mergers (both rotating and non-rotating), and explicitly illustrate the relationship between the discrete hypergraph topology and the continuous Riemannian geometry that is being approximated. Finally, we compare the results produced by this code to the results obtained by means of pure Wolfram model evolution (without the underlying PDE system), using a hypergraph substitution rule that provably satisfies the Einstein field equations in the continuum limit.

gr-qc↗

arXiv2021

Some Relativistic and Gravitational Properties of the Wolfram Model

The Wolfram Model, which is a slight generalization of the model first introduced by Stephen Wolfram in A New Kind of Science (NKS), is a discrete spacetime formalism in which space is represented by a hypergraph whose dynamics are determined by abstract replacement operations on set systems, and in which the conformal structure of spacetime is represented by a causal graph. The purpose of this article is to present rigorous mathematical derivations of many key properties of such models in the continuum limit, as first discussed in NKS, including the fact that large classes of them obey discrete forms of both special and general relativity. First, we prove that causal invariance (namely, the requirement that all causal graphs be isomorphic, irrespective of the choice of hypergraph updating order) is equivalent to a discrete version of general covariance, with changes to the updating order corresponding to discrete gauge transformations. This fact then allows one to deduce a discrete analog of Lorentz covariance, and the resultant physical consequences of discrete Lorentz transformations. We also introduce discrete notions of Riemann and Ricci curvature for hypergraphs, and prove that the correction factor for the volume of a discrete spacetime cone in a causal graph corresponding to curved spacetime of fixed dimensionality is proportional to a timelike projection of the discrete spacetime Ricci tensor, subsequently using this fact (along with the assumption that the updating rules preserve the dimensionality of the causal graph in limiting cases) to prove that the most general set of constraints on the discrete spacetime Ricci tensor corresponds to a discrete form of the Einstein field equations.

cs.DM↗