Abstract We present the application of a class of deep learning, known as Physics Informed Neural Networks (PINN), to inversion and surrogate modeling in solid mechanics. We explain how to incorporate the momentum balance and constitutive relations into PINN, and explore in detail the application to linear elasticity, and illustrate its extension to nonlinear problems through an example that showcases von Mises elastoplasticity. While common PINN algorithms are based on training one deep neural network (DNN), we propose a multi-network model that results in more accurate representation of the field variables. To validate the model, we test the framework on synthetic data generated from analytical and numerical reference solutions. We study convergence of the PINN model, and show that Isogeometric Analysis (IGA) results in superior accuracy and convergence characteristics compared with classic low-order Finite Element Method (FEM). We also show the applicability of the framework for transfer learning, and find vastly accelerated convergence during network re-training. Finally, we find that honoring the physics leads to improved robustness: when trained only on a few parameters, we find that the PINN model can accurately predict the solution for a wide range of parameters new to the network—thus pointing to an important application of this framework to sensitivity analysis and surrogate modeling.
Introduction Foreward by Tudor Ratiu and Richard Cushman Preliminaries Differential Theory Calculus on Manifolds Analytical Dynamics Hamiltonian and Lagrangian Systems Hamiltonian Systems with Symmetry Hamiltonian-Jacobi Theory and Mathematical Physics An Outline of Qualitative Dynamics Topological Dynamics Differentiable Dynamics Hamiltonian Dynamics Celestial Mechanics The Two-Body Problem The Three-Body Problem.
Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degree, analytical methods are usually not enough. One has to resort to discretization methods. For engineering problems, probably the best known option is the finite element method (FEM). However, powerful alternatives such as mesh-free methods and Isogeometric Analysis (IGA) are also available. The fundamental idea is to approximate the solution of the PDE by means of functions specifically built to have some desirable properties. In this contribution, we explore Deep Neural Networks (DNNs) as an option for approximation. They have shown impressive results in areas such as visual recognition. DNNs are regarded here as function approximation machines. There is great flexibility to define their structure and important advances in the architecture and the efficiency of the algorithms to implement them make DNNs a very interesting alternative to approximate the solution of a PDE. We concentrate in applications that have an interest for Computational Mechanics. Most contributions that have decided to explore this possibility have adopted a collocation strategy. In this contribution, we concentrate in mechanical problems and analyze the energetic format of the PDE. The energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem. As proofs of concept, we deal with several problems and explore the capabilities of the method for applications in engineering.
This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, the fluctuation–dissipation relation, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. In some cases, it is possible to extend their regimes of applicability beyond the standard thermal equilibrium domain. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We present results from numerical experiments and analytical studies of quantum quenches at integrability. We introduce the concept of the generalized Gibbs ensemble and discuss its connection with ideas of prethermalization in weakly interacting systems.
The plant cell wall is a dynamic network of several biopolymers and structural proteins including cellulose, pectin, hemicellulose and lignin. Cellulose is one of the main load bearing components of this complex, heterogeneous structure, and in this way, is an important regulator of cell wall growth and mechanics. Glucan chains of cellulose aggregate via hydrogen bonds and van der Waals forces to form long thread-like crystalline structures called cellulose microfibrils. The shape, size, and crystallinity of these microfibrils are important structural parameters that influence mechanical properties of the cell wall and these parameters are likely important determinants of cell wall digestibility for biofuel conversion. Cellulose–cellulose and cellulose-matrix interactions also contribute to the regulation of the mechanics and growth of the cell wall. As a consequence, much emphasis has been placed on extracting valuable structural details about cell wall components from several techniques, either individually or in combination, including diffraction/scattering, microscopy, and spectroscopy. In this review, we describe efforts to characterize the organization of cellulose in plant cell walls. X-ray scattering reveals the size and orientation of microfibrils; diffraction reveals unit lattice parameters and crystallinity. The presence of different cell wall components, their physical and chemical states, and their alignment and orientation have been identified by Infrared, Raman, Nuclear Magnetic Resonance, and Sum Frequency Generation spectroscopy. Direct visualization of cell wall components, their network-like structure, and interactions between different components has also been made possible through a host of microscopic imaging techniques including scanning electron microscopy, transmission electron microscopy, and atomic force microscopy. This review highlights advantages and limitations of different analytical techniques for characterizing cellulose structure and its interaction with other wall polymers. We also delineate emerging opportunities for future developments of structural characterization tools and multi-modal analyses of cellulose and plant cell walls. Ultimately, elucidation of the structure of plant cell walls across multiple length scales will be imperative for establishing structure-property relationships to link cell wall structure to control of growth and mechanics.
Xing Liu, Christos E. Athanasiou, N. Padture
et al.
Abstract Analytical and empirical solutions to engineering problems are usually preferred because of their convenience in applications. However, they are not always accessible in complex problems. A new class of solutions, based on machine learning (ML) models such as regression trees and neural networks (NNs), are proposed and their feasibility and value are demonstrated through the analysis of fracture toughness measurements. It is found that both solutions based on regression trees and NNs can provide accurate results for the specific problem, but NN-based solutions outperform regression-tree-based solutions in terms of their simplicity. This example demonstrates that ML solutions are a major improvement over analytical and empirical solutions in terms of both reliable functionality and rapid deployment. When analytical solutions are not available, the use of ML solutions can overcome the limitations of empirical solutions and substantially change the way that engineering problems are solved.
We present an approach for using machine learning to automatically discover the governing equations and unknown properties (in this case, masses) of real physical systems from observations. We train a ‘graph neural network’ to simulate the dynamics of our Solar System’s Sun, planets, and large moons from 30 years of trajectory data. We then use symbolic regression to correctly infer an analytical expression for the force law implicitly learned by the neural network, which our results showed is equivalent to Newton’s law of gravitation. The key assumptions our method makes are translational and rotational equivariance, and Newton’s second and third laws of motion. It did not, however, require any assumptions about the masses of planets and moons or physical constants, but nonetheless, they, too, were accurately inferred with our method. Naturally, the classical law of gravitation has been known since Isaac Newton, but our results demonstrate that our method can discover unknown laws and hidden properties from observed data.
Thomas W. Grimm, Giovanni Ravazzini, Mick van Vliet
Abstract Many observables in quantum field theories are involved non-analytic functions of the parameters of the theory. However, it is expected that they are not arbitrarily wild, but rather have only a finite amount of geometric complexity. This expectation has been recently formalized by a tameness principle: physical observables should be definable in o-minimal structures and their sharp refinements. In this work, we show that a broad class of non-analytic partition and correlation functions are tame functions in the o-minimal structure known as ℝ G $$ {\mathbb{R}}_{\mathcal{G}} $$ — the structure defining Gevrey functions. Using a perturbative approach, we expand the observables in asymptotic series in powers of a small coupling constant. Although these series are often divergent, they can be Borel-resummed in the absence of Stokes phenomena to yield the full partition and correlation functions. We show that this makes them definable in ℝ G $$ {\mathbb{R}}_{\mathcal{G}} $$ and provide a number of motivating examples. These include certain 0-dimensional quantum field theories and a set of higher-dimensional quantum field theories that can be analyzed using constructive field theory. Finally, we discuss how the eigenvalues of certain Hamiltonians in quantum mechanics are also definable in ℝ G $$ {\mathbb{R}}_{\mathcal{G}} $$ .
Nuclear and particle physics. Atomic energy. Radioactivity
Classical Electrodynamics in ponderable media remains defined by a century-long debate over force and energy localization. While the prevailing view treats competing formulations (Minkowski, Abraham, etc.) as equivalent conventions, this monograph argues that global conservation is insufficient for physical validity. A formulation must be mechanically coherent: power transfer must strictly equal the work rate of the force acting on the mass target. We formalize this requirement as the Force--Energy Consistency Criterion (FECC) -- a ``Kinematic Lock'' ($ P = \mathbf{f} \cdot \mathbf{v} $) -- and use it to audit standard macroscopic tensors. The analysis demonstrates that only the Macroscopic Vacuum (Lorentz) Formulation offers a mechanically consistent description of total energy-momentum transfer. The internal distribution of this energy is shown to be macroscopically indeterminate. By reinterpreting spatial averaging as spectral filtering, we reconstruct the theory from the microscopic baseline. This perspective identifies a universal host interface that routes electromagnetic energy into mechanical work, heat, and reversible storage, revealing a structural isomorphism where thermodynamics, mechanics, and electrodynamics emerge as coupled spectral projections.
We study the statistical properties of active Ornstein-Uhlenbeck particles (AOUPs). In this simplest of models, the Gaussian white noise of overdamped Brownian colloids is replaced by a Gaussian colored noise. This suffices to grant this system the hallmark properties of active matter, while still allowing for analytical progress. We study in detail the steady-state distribution of AOUPs in the small persistence time limit and for spatially varying activity. At the collective level, we show AOUPs to experience motility-induced phase separation both in the presence of pairwise forces or due to quorum-sensing interactions. We characterize both the instability mechanism leading to phase separation and the resulting phase coexistence. We probe how, in the stationary state, AOUPs depart from their thermal equilibrium limit by investigating the emergence of ratchet currents and entropy production. In the small persistence time limit, we show how fluctuation-dissipation relations are recovered. Finally, we discuss how the emerging properties of AOUPs can be characterized from the dynamics of their collective modes.
Professor Dulat Syzdykbekovich Dzhumabaev, Doctor of Physical and Mathematical Sciences, was a prominent scientist, a well-known specialist in the field of the qualitative theory of differential and integro-differential equations, the theory of nonlinear operator equations, numerical and approximate methods for solving boundary value problems.
The paper gathers and unifies mechanical stability conditions for all symmetry classes of 3D and 2D materials under arbitrary load. The methodology is based on the spectral decomposition of the fourth-order stiffness tensors mapped to second-order tensors using orthonormal (Mandel) notation, and the verification of the positivity of the so-called Kelvin moduli. An explicit set of stability conditions for 3D and 2D crystals of higher symmetry is also included, as well as a Mathematica notebook that allows mechanical stability analysis for crystals, stress-free and stressed, of arbitrary symmetry under arbitrary loads.
The mechanism governing the evolution of controlled quantum systems is often obscured, making their dynamics hard to interpret. Mitra and Rabitz {[Phys. Rev. A 67, 033407 (2003)]} define mechanism via a perturbative expansion of pathways between eigenstates; the evolution of the system is driven by the constructive and destructive interference of these pathway amplitudes. In this paper, we explore mechanism in controlled single-qubit systems and describe novel analytic methods for computing the mechanism underlying the evolution of a single qubit under a piecewise constant control.
The aim of this paper is to explore the relationship between invariant cones and nonlinear normal modes in piecewise linear mechanical systems. As a key result, we extend the invariant cone concept, originally established for homogeneous piecewise linear systems, to a class of inhomogeneous continuous piecewise linear systems. The inhomogeneous terms can be constant and/or time-dependent, modeling nonsmooth mechanical systems with a clearance gap and external harmonic forcing, respectively. Using an augmented state vector, a modified invariant cone problem is formulated and solved to compute the nonlinear normal modes, understood as periodic solutions of the underlying conservative dynamics. An important contribution is that invariant cones of the underlying homogeneous system can be regarded as a singularity in the theory of nonlinear normal modes of continuous piecewise linear systems. In addition, we use a similar methodology to take external harmonic forcing into account. We illustrate our approach using numerical examples of mechanical oscillators with a unilateral elastic contact. The resulting backbone curves and frequency response diagrams are compared to the results obtained using the shooting method and brute force time integration.
Abstract The cracked chevron notched Brazilian disc (CCNBD) specimen is a suggested testing method to measure Mode-I fracture toughness of rocks by ISRM, which is widely adopted in the laboratory experiments. However, sizes of CCNBD rock specimens are uncertain in the laboratory experiments, which leads to be inaccurate in measurement of Mode-I fracture toughness of rocks in tests. In this work, four machine learning approaches, including decision regression tree, random regression forest, extra regression tree and fully-connected neural networks (FCNNs) are developed and their feasibility and value are demonstrated through the analysis and predictions of Mode-I fracture toughness of rocks. It can be found that solutions based on the four machine learning approaches can provide the accurate results for predicting Mode-I fracture toughness of rock by in ISRM-suggested CCNBD rock specimens. The random regression forest is more suitable to predict Mode-I fracture toughness of rocks in ISRM-suggested CCNBD rock tests than others. The reliable functionality and rapid development of machine learning solutions are demonstrated that it is a major improvement over the previous analytical and empirical solutions by this example. When analytical and empirical solutions are not available, machine learning approaches overcome the associated limitations, which provides a substantially way to solve rock engineering problems.
Random quantum circuits are proficient information scramblers and efficient generators of randomness, rapidly approximating moments of the unitary group. We study the convergence of local random quantum circuits to unitary $k$-designs. Employing a statistical mechanical mapping, we give an exact expression of the distance to forming an approximate design as a lattice partition function. In the statistical mechanics model, the approach to randomness has a simple interpretation in terms of domain walls extending through the circuit. We analytically compute the second moment, showing that random circuits acting on $n$ qudits form approximate 2-designs in $O(n)$ depth, as is known. Furthermore, we argue that random circuits form approximate unitary $k$-designs in $O(nk)$ depth and are thus essentially optimal in both $n$ and $k$. We can show this in the limit of large local dimension, but more generally rely on a conjecture about the dominance of certain domain wall configurations.