We propose a framework to solve non-linear and history-dependent mechanical problems based on a hybrid classical computer-quantum annealer approach. Quantum Computers are anticipated to solve particular operations exponentially faster. The available possible operations are however not as versatile as with a classical computer. However, quantum annealers (QAs) is well suited to evaluate the minimum state of a Hamiltonian quadratic potential. Therefore, we reformulate the elasto-plastic finite element problem as a double minimisation process framed at the structural scale using the variational updates formulation. In order to comply with the expected quadratic nature of the Hamiltonian, the resulting non-linear minimisation problems are iteratively solved with the suggested Quantum Annealing-assisted Sequential Quadratic Programming (QA-SQP): a sequence of minimising quadratic problems is performed by approximating the objective function by a quadratic Taylor's series. Each quadratic minimisation problem of continuous variables is then transformed into a binary quadratic problem. This binary quadratic minimisation problem can be solved on quantum annealing hardware such as the D-Wave system. The applicability of the proposed framework is demonstrated with one and two-dimensional elasto-plastic numerical benchmarks. The current work provides a pathway of performing general non-linear finite element simulations assisted by quantum computing.
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Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling, due to their state-of-the art performance in many generation tasks while relying on mathematical foundations such as stochastic differential equations (SDEs) and ordinary differential equations (ODEs). Empirically, it has been reported that ODE based samples are inferior to SDE based samples. In this paper we rigorously describe the range of dynamics and approximations that arise when training score-based diffusion models, including the true SDE dynamics, the neural approximations, the various approximate particle dynamics that result, as well as their associated Fokker--Planck equations and the neural network approximations of these Fokker--Planck equations. We systematically analyse the difference between the ODE and SDE dynamics of score-based diffusion models, and link it to an associated Fokker--Planck equation. We derive a theoretical upper bound on the Wasserstein 2-distance between the ODE- and SDE-induced distributions in terms of a Fokker--Planck residual. We also show numerically that conventional score-based diffusion models can exhibit significant differences between ODE- and SDE-induced distributions which we demonstrate using explicit comparisons. Moreover, we show numerically that reducing the Fokker--Planck residual by adding it as an additional regularisation term leads to closing the gap between ODE- and SDE-induced distributions. Our experiments suggest that this regularisation can improve the distribution generated by the ODE, however that this can come at the cost of degraded SDE sample quality.
In this work, a class of continuous-time autonomous dynamical systems describing many important phenomena and processes arising in real-world applications is considered. We apply the nonstandard finite difference (NSFD) methodology proposed by Mickens to design a generalized NSFD method for the dynamical system models under consideration. This method is constructed based on a novel non-local approximation for the right-side functions of the dynamical systems. It is proved by rigorous mathematical analyses that the NSFD method is dynamically consistent with respect to positivity, asymptotic stability and three classes of conservation laws, including direct conservation, generalized conservation and sub-conservation laws. Furthermore, the NSFD method is easy to be implemented and can be applied to solve a broad range of mathematical models arising in real-life. Finally, a set of numerical experiments is performed to illustrate the theoretical findings and to show advantages of the proposed NSFD method.
This paper introduces a novel approach to approximate a broad range of reaction-convection-diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves accuracy of $O(h^k)$ in the energy norm, where $k$ represents the underlying polynomial degree. To validate the approach, a series of numerical experiments is conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favourable performance of the current approach.
We study the computational problem of rigorously describing the asymptotic behaviour of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit, both as a spatial object (attractor set) and as a statistical distribution (physical measure), and prove upper bounds on the computational resources of computing descriptions of these objects with arbitrary accuracy. We also study how these bounds are affected by different dynamical constrains and provide several examples showing that our bounds are sharp in general. In particular, we exhibit a computable interval map having a unique transitive attractor with Cantor set structure supporting a unique physical measure such that both the attractor and the measure are non computable.
In this paper, we propose a robust low-order stabilization-free virtual element method on quadrilateral meshes for linear elasticity that is based on the stress-hybrid principle. We refer to this approach as the Stress-Hybrid Virtual Element Method (SH-VEM). In this method, the Hellinger$-$Reissner variational principle is adopted, wherein both the equilibrium equations and the strain-displacement relations are variationally enforced. We consider small-strain deformations of linear elastic solids in the compressible and near-incompressible regimes over quadrilateral (convex and nonconvex) meshes. Within an element, the displacement field is approximated as a linear combination of canonical shape functions that are $\textit{virtual}$. The stress field, similar to the stress-hybrid finite element method of Pian and Sumihara, is represented using a linear combination of symmetric tensor polynomials. A 5-parameter expansion of the stress field is used in each element, with stress transformation equations applied on distorted quadrilaterals. In the variational statement of the strain-displacement relations, the divergence theorem is invoked to express the stress coefficients in terms of the nodal displacements. This results in a formulation with solely the nodal displacements as unknowns. Numerical results are presented for several benchmark problems from linear elasticity. We show that SH-VEM is free of volumetric and shear locking, and it converges optimally in the $L^2$ norm and energy seminorm of the displacement field, and in the $L^2$ norm of the hydrostatic stress.
We propose a second order exponential scheme suitable for two-component coupled systems of stiff evolutionary advection--diffusion--reaction equations in two and three space dimensions. It is based on a directional splitting of the involved matrix functions, which allows for a simple yet efficient implementation through the computation of small-sized exponential-like functions and tensor-matrix products. The procedure straightforwardly extends to the case of an arbitrary number of components and to any space dimension. Several numerical examples in 2D and 3D with physically relevant (advective) Schnakenberg, FitzHugh--Nagumo, DIB, and advective Brusselator models clearly show the advantage of the approach against state-of-the-art techniques.
Deciphering and integrating invariants for neural operator learning with various physical mechanisms
Neural operators have been explored as surrogate models for simulating physical systems to overcome the limitations of traditional partial differential equation (PDE) solvers. However, most existing operator learning methods assume that the data originate from a single physical mechanism, limiting their applicability and performance in more realistic scenarios. To this end, we propose Physical Invariant Attention Neural Operator (PIANO) to decipher and integrate the physical invariants (PI) for operator learning from the PDE series with various physical mechanisms. PIANO employs self-supervised learning to extract physical knowledge and attention mechanisms to integrate them into dynamic convolutional layers. Compared to existing techniques, PIANO can reduce the relative error by 13.6\%-82.2\% on PDE forecasting tasks across varying coefficients, forces, or boundary conditions. Additionally, varied downstream tasks reveal that the PI embeddings deciphered by PIANO align well with the underlying invariants in the PDE systems, verifying the physical significance of PIANO. The source code will be publicly available at: //github.com/optray/PIANO.
Refinement calculus provides a structured framework for the progressive and modular development of programs, ensuring their correctness throughout the refinement process. This paper introduces a refinement calculus tailored for quantum programs. To this end, we first study the partial correctness of nondeterministic programs within a quantum while language featuring prescription statements. Orthogonal projectors, which are equivalent to subspaces of the state Hilbert space, are taken as assertions for quantum states. In addition to the denotational semantics where a nondeterministic program is associated with a set of trace-nonincreasing super-operators, we also present their semantics in transforming a postcondition to the weakest liberal postconditions and, conversely, transforming a precondition to the strongest postconditions. Subsequently, refinement rules are introduced based on these dual semantics, offering a systematic approach to the incremental development of quantum programs applicable in various contexts. To illustrate the practical application of the refinement calculus, we examine examples such as the implementation of a $Z$-rotation gate, the repetition code, and the quantum-to-quantum Bernoulli factory. Furthermore, we present Quire, a Python-based interactive prototype tool that provides practical support to programmers engaged in the stepwise development of correct quantum programs.
The main reason for query model's prominence in complexity theory and quantum computing is the presence of concrete lower bounding techniques: polynomial and adversary method. There have been considerable efforts to give lower bounds using these methods, and to compare/relate them with other measures based on the decision tree. We explore the value of these lower bounds on quantum query complexity and their relation with other decision tree based complexity measures for the class of symmetric functions, arguably one of the most natural and basic sets of Boolean functions. We show an explicit construction for the dual of the positive adversary method and also of the square root of private coin certificate game complexity for any total symmetric function. This shows that the two values can't be distinguished for any symmetric function. Additionally, we show that the recently introduced measure of spectral sensitivity gives the same value as both positive adversary and approximate degree for every total symmetric Boolean function. Further, we look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. Finally, we study how large certificate complexity and block sensitivity can be as compared to sensitivity for symmetric functions (even up to constant factors). We show tight separations, i.e., give upper bounds on possible separations and construct functions achieving the same.
We propose and compare methods for the analysis of extreme events in complex systems governed by PDEs that involve random parameters, in situations where we are interested in quantifying the probability that a scalar function of the system's solution is above a threshold. If the threshold is large, this probability is small and its accurate estimation is challenging. To tackle this difficulty, we blend theoretical results from large deviation theory (LDT) with numerical tools from PDE-constrained optimization. Our methods first compute parameters that minimize the LDT-rate function over the set of parameters leading to extreme events, using adjoint methods to compute the gradient of this rate function. The minimizers give information about the mechanism of the extreme events as well as estimates of their probability. We then propose a series of methods to refine these estimates, either via importance sampling or geometric approximation of the extreme event sets. Results are formulated for general parameter distributions and detailed expressions are provided when Gaussian distributions. We give theoretical and numerical arguments showing that the performance of our methods is insensitive to the extremeness of the events we are interested in. We illustrate the application of our approach to quantify the probability of extreme tsunami events on shore. Tsunamis are typically caused by a sudden, unpredictable change of the ocean floor elevation during an earthquake. We model this change as a random process, which takes into account the underlying physics. We use the one-dimensional shallow water equation to model tsunamis numerically. In the context of this example, we present a comparison of our methods for extreme event probability estimation, and find which type of ocean floor elevation change leads to the largest tsunamis on shore.
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