Gradient descent globally solves average-case non-resonant physical design problems

• URL: http://arxiv.org/abs/2111.02978v1
• Date: Thu, 4 Nov 2021 16:22:07 GMT
• Title: Gradient descent globally solves average-case non-resonant physical design problems
• Abstract summary: This paper addresses optimisation problems occurring in a wide variety of physical design problems, including but not limited to optical engineering, quantum control, structural engineering. The physics of the state of the system is often the question of the convergence globally optimal design.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Optimization problems occurring in a wide variety of physical design problems, including but not limited to optical engineering, quantum control, structural engineering, involve minimization of a simple cost function of the state of the system (e.g. the optical fields, the quantum state) while being constrained by the physics of the system. The physics constraints often makes such problems non-convex and thus only locally solvable, leaving open the question of finding the globally optimal design. In this paper, I consider design problems whose physics is described by bi-affine equality constraints, and show that under assumptions on the stability of these constraints and the physical system being non-resonant, gradient descent globally solves a typical physical design problem. The key technical contributions of this paper are (i) outline a criteria that ensure the convergence of gradient descent to an approximate global optima in the limit of large problem sizes, and (ii) use random matrix theory to outline ensembles of physically motivated problems which, on an average, satisfy this convergence criteria.

Related papers

• Learned Diffractive Optics for Quantum-Optimal Inference [42.15593888902416]
  Quantum mechanics sets the ultimate bounds on photon-limited sensing.<n>This is particularly the case for visual sensing problems, where the goal is to infer features of a distant object based on the spatial structure of the light field it emits or reflects.<n>Here, we apply learned diffractive optics to state discrimination and parameter estimation of coherent and diffraction-limited incoherent light fields under a restricted photon budget.
  arXiv  Detail & Related papers  (2026-09-07T21:52:13Z)
• Scalable Ground-State Certification of Quantum Spin Systems via Structured Noncommutative Polynomial Optimization [3.4234084547821477]
  A fundamental challenge in quantum physics is determining the ground-state properties of many-body systems.<n>In this article, we demonstrate that leveraging the inherent structures of the system can significantly mitigate these scalability challenges.
  arXiv  Detail & Related papers  (2026-04-02T03:00:01Z)
• Multi-material Multi-physics Topology Optimization with Physics-informed Gaussian Process Priors [5.910614452545977]
  We propose a framework based on physics-informed Gaussian processes (PIGPs)<n>In our approach, the primary, adjoint, and design variables are represented by independent GP priors.<n>We demonstrate the capability of the proposed framework on benchmark TO problems such as compliance minimization, heat conduction optimization, and compliant mechanism design.
  arXiv  Detail & Related papers  (2026-02-19T19:28:18Z)
• Online Inference of Constrained Optimization: Primal-Dual Optimality and Sequential Quadratic Programming [55.848340925419286]
  We study online statistical inference for the solutions of quadratic optimization problems with equality and inequality constraints.<n>We develop a sequential programming (SSQP) method to solve these problems, where the step direction is computed by sequentially performing an approximation of the objective and a linear approximation of the constraints.<n>We show that our method global almost moving-average convergence and exhibits local normality with an optimal primal-dual limiting matrix in the sense of Hjek and Le Cam.
  arXiv  Detail & Related papers  (2025-11-27T06:16:17Z)
• Physics-informed neural network solves minimal surfaces in curved spacetime [0.0]
  We develop a framework based on physics-informed neural networks (PINNs) for solving boundary value problems involving minimal surfaces in curved spacetimes.<n>Our approach enables robust and accurate solutions to both ordinary and partial differential equations with complex boundary conditions.
  arXiv  Detail & Related papers  (2025-09-13T15:41:13Z)
• Robust Variational Ground-State Solvers via Dissipative Quantum Feedback Models [3.7346004746366384]
  We propose a variational framework for solving ground-state problems of open quantum systems governed by quantum differential equations.<n>By parameterizing a dissipative quantum optical system, we minimize its steady-state energy to approximate the ground state of a target Hamiltonian.<n>This framework is compatible with experimental platforms such as cavity quantum electrodynamics (QED) and photonic crystal circuits.
  arXiv  Detail & Related papers  (2025-07-26T15:28:35Z)
• PromptCoT: Synthesizing Olympiad-level Problems for Mathematical Reasoning in Large Language Models [59.920971312822736]
  We introduce PromptCoT, a novel approach for automatically generating high-quality Olympiad-level math problems. The proposed method synthesizes complex problems based on mathematical concepts and the rationale behind problem construction. Our method is evaluated on standard benchmarks including GSM8K, MATH-500, and AIME2024, where it consistently outperforms existing problem generation methods.
  arXiv  Detail & Related papers  (2025-03-04T06:32:30Z)
• Exploring the topological sector optimization on quantum computers [5.458469081464264]
  topological sector optimization (TSO) problem attracts particular interests in the quantum many-body physics community. We demonstrate that the optimization difficulties of TSO problem are not restricted to the gaplessness, but are also due to the topological nature. To solve TSO problems, we utilize quantum imaginary time evolution (QITE) with a possible realization on quantum computers.
  arXiv  Detail & Related papers  (2023-10-06T14:51:07Z)
• Quantum algorithms: A survey of applications and end-to-end complexities [90.05272647148196]
  The anticipated applications of quantum computers span across science and industry. We present a survey of several potential application areas of quantum algorithms. We outline the challenges and opportunities in each area in an "end-to-end" fashion.
  arXiv  Detail & Related papers  (2023-10-04T17:53:55Z)
• Physics-Constrained Hardware-Efficient Ansatz on Quantum Computers that is Universal, Systematically Improvable, and Size-consistent [1.1749564892273827]
  We introduce a physics-constrained approach for designing hardware-efficient ansatz (HEA) on quantum computers. Specifically, we require that the target HEA to be universal, systematically improvable, and size-consistent. We present a concrete realization of HEA that satisfies all these fundamental constraints while only requiring linear qubit connectivity.
  arXiv  Detail & Related papers  (2023-07-07T12:52:20Z)
• Many Physical Design Problems are Sparse QCQPs [0.0]
  Physical design applies to a desired objective (e.g. strong light-matter interactions) to quantum equations. computing an optimal design highlights the possibility of developing global optimal design differential equations.
  arXiv  Detail & Related papers  (2023-03-30T20:10:57Z)
• Multiobjective variational quantum optimization for constrained problems: an application to Cash Management [45.82374977939355]
  We introduce a new method for solving optimization problems with challenging constraints using variational quantum algorithms. We test our proposal on a real-world problem with great relevance in finance: the Cash Management problem. Our empirical results show a significant improvement in terms of the cost of the achieved solutions, but especially in the avoidance of local minima.
  arXiv  Detail & Related papers  (2023-02-08T17:09:20Z)
• Optimal Sufficient Requirements on the Embedded Ising Problem in Polynomial Time [0.0]
  We evaluate sufficient requirements on such an embedded Ising problem analytically and transform them into a linear optimization problem. With an objective function aiming to minimize the maximal absolute problem parameter, the precision issues of the annealers are addressed. This allows to formulate provably equivalent embedded Ising problems in a practical setup.
  arXiv  Detail & Related papers  (2023-02-08T16:13:55Z)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• Adiabatic Quantum Computing for Multi Object Tracking [170.8716555363907]
  Multi-Object Tracking (MOT) is most often approached in the tracking-by-detection paradigm, where object detections are associated through time. As these optimization problems are often NP-hard, they can only be solved exactly for small instances on current hardware. We show that our approach is competitive compared with state-of-the-art optimization-based approaches, even when using of-the-shelf integer programming solvers.
  arXiv  Detail & Related papers  (2022-02-17T18:59:20Z)
• Multi-Objective Constrained Optimization for Energy Applications via Tree Ensembles [55.23285485923913]
  Energy systems optimization problems are complex due to strongly non-linear system behavior and multiple competing objectives. In some cases, proposed optimal solutions need to obey explicit input constraints related to physical properties or safety-critical operating conditions. This paper proposes a novel data-driven strategy using tree ensembles for constrained multi-objective optimization of black-box problems.
  arXiv  Detail & Related papers  (2021-11-04T20:18:55Z)
• Physics informed neural networks for continuum micromechanics [68.8204255655161]
  Recently, physics informed neural networks have successfully been applied to a broad variety of problems in applied mathematics and engineering. Due to the global approximation, physics informed neural networks have difficulties in displaying localized effects and strong non-linear solutions by optimization. It is shown, that the domain decomposition approach is able to accurately resolve nonlinear stress, displacement and energy fields in heterogeneous microstructures obtained from real-world $mu$CT-scans.
  arXiv  Detail & Related papers  (2021-10-14T14:05:19Z)
• Offline Model-Based Optimization via Normalized Maximum Likelihood Estimation [101.22379613810881]
  We consider data-driven optimization problems where one must maximize a function given only queries at a fixed set of points. This problem setting emerges in many domains where function evaluation is a complex and expensive process. We propose a tractable approximation that allows us to scale our method to high-capacity neural network models.
  arXiv  Detail & Related papers  (2021-02-16T06:04:27Z)
• Physics-informed neural networks with hard constraints for inverse design [3.8191831921441337]
  We propose a new deep learning method -- physics-informed neural networks with hard constraints (hPINNs) -- for solving topology optimization. We demonstrate the effectiveness of hPINN for a holography problem in optics and a fluid problem of Stokes flow.
  arXiv  Detail & Related papers  (2021-02-09T03:18:15Z)
• Fusion-based quantum computation [43.642915252379815]
  Fusion-based quantum computing (FBQC) is a model of universal quantum computation in which entangling measurements, called fusions, are performed on qubits of small constant-sized entangled resource states. We introduce a stabilizer formalism for analyzing fault tolerance and computation in these schemes. This framework naturally captures the error structure that arises in certain physical systems for quantum computing, such as photonics.
  arXiv  Detail & Related papers  (2021-01-22T20:00:22Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.