Alex H. Barnett, Gary Marple, Shravan Veerapaneni et al.
We present a spectrally-accurate scheme to turn a boundary integral formulation for an elliptic PDE on a single unit cell geometry into one for the fully periodic problem. Applications include computing the effective permeability of composite media (homogenization), and microfluidic chip design. Our basic idea is to exploit a small least squares solve to apply periodicity without ever handling periodic Green's functions. We exhibit fast solvers for the two-dimensional (2D) doubly-periodic Neumann Laplace problem (flow around insulators), and Stokes non-slip fluid flow problem, that for inclusions with smooth boundaries achieve 12-digit accuracy, and can handle thousands of inclusions per unit cell. We split the infinite sum over the lattice of images into a directly-summed "near" part plus a small number of auxiliary sources which represent the (smooth) remaining "far" contribution. Applying physical boundary conditions on the unit cell walls gives an expanded linear system, which, after a rank-1 or rank-3 correction and a Schur complement, leaves a well-conditioned square system which can be solved iteratively using fast multipole acceleration plus a low-rank term. We are rather explicit about the consistency and nullspaces of both the continuous and discretized problems. The scheme is simple (no lattice sums, Ewald methods, nor particle meshes are required), allows adaptivity, and is essentially dimension- and PDE-independent, so would generalize without fuss to 3D and to other non-oscillatory elliptic problems such as elastostatics. We incorporate recently developed spectral quadratures that accurately handle close-to-touching geometries. We include many numerical examples, and provide a software implementation.
Variational optimization of neural-network representations of quantum states has been successfully applied to solve interacting fermionic problems. Despite rapid developments, significant scalability challenges arise when considering molecules of large scale, which correspond to non-locally interacting quantum spin Hamiltonians consisting of sums of thousands or even millions of Pauli operators. In this work, we introduce scalable parallelization strategies to improve neural-network-based variational quantum Monte Carlo calculations for ab-initio quantum chemistry applications. We establish GPU-supported local energy parallelism to compute the optimization objective for Hamiltonians of potentially complex molecules. Using autoregressive sampling techniques, we demonstrate systematic improvement in wall-clock timings required to achieve CCSD baseline target energies. The performance is further enhanced by accommodating the structure of resultant spin Hamiltonians into the autoregressive sampling ordering. The algorithm achieves promising performance in comparison with the classical approximate methods and exhibits both running time and scalability advantages over existing neural-network based methods.
Eduardo Corona, David Gorsich, Paramsothy Jayakumar et al.
In the last two decades, increased need for high-fidelity simulations of the time evolution and propagation of forces in granular media has spurred renewed interest in discrete element method (DEM) modeling of frictional contact. Force penalty methods, while economic and accessible, introduce artificial stiffness, requiring small time steps to retain numerical stability. Optimization-based methods, which enforce contacts geometrically through complementarity constraints, allow the use of larger time steps at the expense of solving a nonlinear complementarity problem (NCP) each time step. We review the latest efforts to produce solvers for this NCP, focusing on its relaxation to a cone complementarity problem (CCP) and solution via an equivalent quadratic optimization problem with conic constraints. We distinguish between linearly convergent first order methods and second order methods, which gain quadratic convergence and more robust performance at the expense of the solution of large sparse linear systems. We propose a novel acceleration for the solution of Newton step linear systems in second order methods using low-rank compression based fast direct solvers. We use the Quantized Tensor Train (QTT) decomposition to produce efficient approximate representations of the system matrix and its inverse. This provides a robust framework to accelerate its solution in a direct or a preconditioned iterative method. In a number of numerical tests, we demonstrate that this approach displays sublinear scaling of precomputation costs, may be efficiently updated across Newton iterations as well as across time steps, and leads to a fast, optimal complexity solution of the Newton step. This allows our method to gain an order of magnitude speedups over state-of-the-art preconditioning techniques for moderate to large-scale systems, mitigating the computational bottleneck of second order methods.
This article aims to summarize recent and ongoing efforts to simulate continuous-variable quantum systems using flow-based variational quantum Monte Carlo techniques, focusing for pedagogical purposes on the example of bosons in the field amplitude (quadrature) basis. Particular emphasis is placed on the variational real- and imaginary-time evolution problems, carefully reviewing the stochastic estimation of the time-dependent variational principles and their relationship with information geometry. Some practical instructions are provided to guide the implementation of a PyTorch code. The review is intended to be accessible to researchers interested in machine learning and quantum information science.
Variational quantum algorithms (VQAs) utilize a hybrid quantum-classical architecture to recast problems of high-dimensional linear algebra as ones of stochastic optimization. Despite the promise of leveraging near- to intermediate-term quantum resources to accelerate this task, the computational advantage of VQAs over wholly classical algorithms has not been firmly established. For instance, while the variational quantum eigensolver (VQE) has been developed to approximate low-lying eigenmodes of high-dimensional sparse linear operators, analogous classical optimization algorithms exist in the variational Monte Carlo (VMC) literature, utilizing neural networks in place of quantum circuits to represent quantum states. In this paper we ask if classical stochastic optimization algorithms can be constructed paralleling other VQAs, focusing on the example of the variational quantum linear solver (VQLS). We find that such a construction can be applied to the VQLS, yielding a paradigm that could theoretically extend to other VQAs of similar form.
Marc Bonnet, Kausik Das, Shravan Veerapaneni et al.
This article presents a computational framework for determining the optimal slip velocity of a microswimmer with arbitrary three-dimensional geometry suspended in a viscous fluid. The objective is to minimize the hydrodynamic power dissipation required to maintain unit speed along the net swimming direction. By exploiting the linearity of the Stokes equations and the Lorentz reciprocal theorem, we derive an explicit linear operator that maps the tangential surface slip velocity to the resulting rigid-body translational and rotational velocities, effectively decoupling the hydrodynamic boundary value problem from the optimization loop. The a priori infinite-dimensional search space for the slip optimization is reduced to the finite dimension $r$ of rigid-body motions by finding an appropriate subspace of the operator's domain. This reduces the PDE-constrained optimization to a low-dimensional programming problem that can be solved at negligible computational cost once the system matrices are assembled. The optimization algorithm requires 2$r$ auxiliary flow problems that are solved numerically using a high-order boundary integral method. We validate the accuracy of the proposed method and present optimal slip profiles and swimming trajectories for a variety of microswimmer shapes. We investigate the effect of some common geometrical symmetries of the swimmer shape on the resulting optimal motion, and in particular present a modified version of the slip optimization algorithm for axisymmetric shapes, where tangential rigid-body velocities may occur
Neural-network quantum states (NQS) has emerged as a powerful application of quantum-inspired deep learning for variational Monte Carlo methods, offering a competitive alternative to existing techniques for identifying ground states of quantum problems. A significant advancement toward improving the practical scalability of NQS has been the incorporation of autoregressive models, most recently transformers, as variational ansatze. Transformers learn sequence information with greater expressiveness than recurrent models, but at the cost of increased time complexity with respect to sequence length. We explore the use of the retentive network (RetNet), a recurrent alternative to transformers, as an ansatz for solving electronic ground state problems in $\textit{ab initio}$ quantum chemistry. Unlike transformers, RetNets overcome this time complexity bottleneck by processing data in parallel during training, and recurrently during inference. We give a simple computational cost estimate of the RetNet and directly compare it with similar estimates for transformers, establishing a clear threshold ratio of problem-to-model size past which the RetNet's time complexity outperforms that of the transformer. Though this efficiency can comes at the expense of decreased expressiveness relative to the transformer, we overcome this gap through training strategies that leverage the autoregressive structure of the model -- namely, variational neural annealing. Our findings support the RetNet as a means of improving the time complexity of NQS without sacrificing accuracy. We provide further evidence that the ablative improvements of neural annealing extend beyond the RetNet architecture, suggesting it would serve as an effective general training strategy for autoregressive NQS.
Oliver Knitter, Dan Zhao, Stefan Leichenauer et al.
Scaling laws have been used to describe how large language model (LLM) performance scales with model size, training data size, or amount of computational resources. Motivated by the fact that neural quantum states (NQS) has increasingly adopted LLM-based components, we seek to understand NQS scaling laws, thereby shedding light on the scalability and optimal performance--resource trade-offs of NQS ansatze. In particular, we identify scaling laws that predict the performance, as measured by absolute error and V-score, for transformer-based NQS as a function of problem size in second-quantized quantum chemistry applications. By performing analogous compute-constrained optimization of the obtained parametric curves, we find that the relationship between model size and training time is highly dependent on loss metric and ansatz, and does not follow the approximately linear relationship found for language models.
Variational quantum algorithms (VQAs) are promising hybrid quantum-classical methods designed to leverage the computational advantages of quantum computing while mitigating the limitations of current noisy intermediate-scale quantum (NISQ) hardware. Although VQAs have been demonstrated as proofs of concept, their practical utility in solving real-world problems -- and whether quantum-inspired classical algorithms can match their performance -- remains an open question. We present a novel application of the variational quantum linear solver (VQLS) and its classical neural quantum states-based counterpart, the variational neural linear solver (VNLS), as key components within a minimum map Newton solver for a complementarity-based rigid body contact model. We demonstrate using the VNLS that our solver accurately simulates the dynamics of rigid spherical bodies during collision events. These results suggest that quantum and quantum-inspired linear algebra algorithms can serve as viable alternatives to standard linear algebra solvers for modeling certain physical systems.
James Stokes, Saibal De, Shravan Veerapaneni et al.
We initiate the study of neural-network quantum state algorithms for analyzing continuous-variable lattice quantum systems in first quantization. A simple family of continuous-variable trial wavefunctons is introduced which naturally generalizes the restricted Boltzmann machine (RBM) wavefunction introduced for analyzing quantum spin systems. By virtue of its simplicity, the same variational Monte Carlo training algorithms that have been developed for ground state determination and time evolution of spin systems have natural analogues in the continuum. We offer a proof of principle demonstration in the context of ground state determination of a stoquastic quantum rotor Hamiltonian. Results are compared against those obtained from partial differential equation (PDE) based scalable eigensolvers. This study serves as a benchmark against which future investigation of continuous-variable neural quantum states can be compared, and points to the need to consider deep network architectures and more sophisticated training algorithms.
Tianchen Zhao, James Stokes, Oliver Knitter et al.
An identification is found between meta-learning and the problem of determining the ground state of a randomly generated Hamiltonian drawn from a known ensemble. A model-agnostic meta-learning approach is proposed to solve the associated learning problem and a preliminary experimental study of random Max-Cut problems indicates that the resulting Meta Variational Monte Carlo accelerates training and improves convergence.
Gary R. Marple, David Gorsich, Paramsothy Jayakumar et al.
A mobility map, which provides maximum achievable speed on a given terrain, is essential for path planning of autonomous ground vehicles in off-road settings. While physics-based simulations play a central role in creating next-generation, high-fidelity mobility maps, they are cumbersome and expensive. For instance, a typical simulation can take weeks to run on a supercomputer and each map requires thousands of such simulations. Recent work at the U.S. Army CCDC Ground Vehicle Systems Center has shown that trained machine learning classifiers can greatly improve the efficiency of this process. However, deciding which simulations to run in order to train the classifier efficiently is still an open problem. According to PAC learning theory, data that can be separated by a classifier is expected to require $\mathcal{O}(1/ε)$ randomly selected points (simulations) to train the classifier with error less than $ε$. In this paper, building on existing algorithms, we introduce an active learning paradigm that substantially reduces the number of simulations needed to train a machine learning classifier without sacrificing accuracy. Experimental results suggest that our sampling algorithm can train a neural network, with higher accuracy, using less than half the number of simulations when compared to random sampling.
Gary Marple, Alex Barnett, Adrianna Gillman et al.
This paper presents a new boundary integral equation (BIE) method for simulating particulate and multiphase flows through periodic channels of arbitrary smooth shape in two dimensions. The authors consider a particular system---multiple vesicles suspended in a periodic channel of arbitrary shape---to describe the numerical method and test its performance. Rather than relying on the periodic Green's function as classical BIE methods do, the method combines the free-space Green's function with a small auxiliary basis, and imposes periodicity as an extra linear condition. As a result, we can exploit existing free-space solver libraries, quadratures, and fast algorithms, and handle a large number of vesicles in a geometrically complex channel. Spectral accuracy in space is achieved using the periodic trapezoid rule and product quadratures, while a first-order semi-implicit scheme evolves particles by treating the vesicle-channel interactions explicitly. New constraint-correction formulas are introduced that preserve reduced areas of vesicles, independent of the number of time steps taken. By using two types of fast algorithms, (i) the fast multipole method (FMM) for the computation of the vesicle-vesicle and the vesicle-channel hydrodynamic interaction, and (ii) a fast direct solver for the BIE on the fixed channel geometry, the computational cost is reduced to $O(N)$ per time step where $N$ is the spatial discretization size. Moreover, the direct solver inverts the wall BIE operator at $t = 0$, stores its compressed representation and applies it at every time step to evolve the vesicle positions, leading to dramatic cost savings compared to classical approaches. Numerical experiments illustrate that a simulation with $N=128, 000$ can be evolved in less than a minute per time step on a laptop.
Dense particulate flow simulations using integral equation methods demand accurate evaluation of Stokes layer potentials on arbitrarily close interfaces. In this paper, we generalize techniques for close evaluation of Laplace double-layer potentials in J. Helsing and R. Ojala, J. Comput. Phys. 227 (2008) 2899-2921. We create a "globally compensated" trapezoid rule quadrature for the Laplace single-layer potential on the interior and exterior of smooth curves. This exploits a complex representation, a product quadrature (in the style of Kress) for the sawtooth function, careful attention to branch cuts, and second-kind barycentric-type formulae for Cauchy integrals and their derivatives. Upon this we build accurate single- and double-layer Stokes potential evaluators by expressing them in terms of Laplace potentials. We test their convergence for vesicle-vesicle interactions, for an extensive set of Laplace and Stokes problems, and when applying the system matrix in a boundary value problem solver in the exterior of multiple close-to-touching ellipses. We achieve typically 12 digits of accuracy using very small numbers of discretization nodes per curve. We provide documented codes for other researchers to use.