Conservation Laws Research Articles

Towards robust data-driven automated recovery of symbolic conservation laws from limited data

Abstract Conservation laws are an inherent feature in many systems modeling real world phenomena, in particular, those modeling biological and chemical systems. If the form of the underlying dynamical system is known, linear algebra and algebraic geometry methods can be used to identify the conservation laws. Our work focuses on using data-driven methods to identify the conservation law(s) in the absence of the knowledge of system dynamics. We develop a robust data-driven computational framework that automates the process of identifying the number and type of the conservation law(s) while keeping the amount of required data to a minimum. We demonstrate that due to relative stability of singular vectors to noise we are able to reconstruct correct conservation laws without the need for excessive parameter tuning. While we focus primarily on biological examples, the framework proposed herein is suitable for a variety of data science applications and can be coupled with other machine learning approaches.

A relativistic scalar model for fractional interaction between dark matter and gravity

Abstract In a series of recent papers we put forward a “fractional gravity”
framework striking an intermediate course between a modified gravity theory and an
exotic dark matter (DM) scenario, which envisages the DM component in virialized
halos to feel a non-local interaction mediated by gravity. The remarkable success
of this model in reproducing several aspects of DM phenomenology motivates us to
look for a general relativistic extension. Specifically, we propose a theory, dubbed
Relativistic Scalar Fractional Gravity or RSFG, in which the trace of the DM stressenergy tensor couples to the scalar curvature via a non-local operator constructed with
a fractional power of the d’Alembertian. We derive the field equations starting from
an action principle, and then we investigate their weak field limit, demonstrating that
in the Newtonian approximation the fractional gravity setup of our previous works is
recovered. We compute the first-order post-Newtonian parameter γ and its relation
with weak lensing, showing that although in RSFG the former deviates from its GR
values of unity, the latter is unaffected. We also perform a standard scalar-vectortensor-decomposition of RSFG in the weak field limit, to highlight that gravitational
waves propagate at the speed of light, though also an additional scalar mode becomes
dynamical. Finally, we derive the modified conservation laws of the DM stress energy
tensor in RSFG, showing that a new non-local force emerges, and hence that the DM
fluid deviates from the geodesic solutions of the field equations.

Comparison performance study of singly-fed and doubly-fed induction generators-based bond-graph wind turbines model

This paper consecrates to a comparative performance study of singly-fed and doubly-fed of induction generators thrusted by wind power turbine of similar generation capacity of 2.5 kW, and constant or variably wind speed. The singly-fed induction generator model could be represented using natural reference frame and doubly-fed induction generator model is described using a Park reference frame. Because of several physical domains existing in both induction generators like mechanical and electrical, modeling of generators is difficult, therefore the modeling based on physical methods takes a high credibility under these conditions. Among the procedures is Bond-graph method that models the systems based on law of mass conservation and/or law of energy conservation containing in the systems. Modeling the parts of both singly-fed and double-fed induction generators are based on Bond-graph method. We found that the impact of stator coil winding for on the current is in the form of changes of its value and steady state intervals; increasing the number of stator coil windings may also lead to increased stator current and the longer their steady state interval. Our study demonstrates also that doubly-fed induction generator possesses advantages compared to singly-fed induction generator, namely better current quality output and an adaptation to fluctuating wind speeds. The performance study is done in constant and variably wind speeds using simulated results of 20-Sim software.

Invariant analysis, invariant subspace method and conservation laws of the (2+1)-dimensional mixed fractional Broer-Kaup-Kupershmidt system

In this paper, we investigate the (2+1)-dimensional mixed fractional Broer-Kaup-Kupershmidt system (MFBKKS) with Riemann–Liouville time fractional derivative and integer order y-derivative. This system models dispersive and nonlinear long gravity waves propagating in shallow water in two horizontal directions with time memories. Lie symmetry analysis and invariant subspace method are distinctly employed to construct the exact solutions to the MFBKKS. Initially, the Lie algebras admitted by MFBKKS are obtained with the help of Lie symmetry analysis. Then, we establish the commutative table, adjoint relations and adjoint transformation matrix. Specifically, the one-dimensional optimal system is established correspondingly, and symmetry reduction is performed. In particular, (2+1)-dimensional MFBKKS is reduced to (1+1)-dimensional mixed fractional system using Erdélyi-Kober fractional differential operator. Further, the power series solution of MFBKKS is constructed via power series method. By applying invariant subspace method, we obtain more exact solutions of MFBKKS. In addition, the conservation laws are derived by new Noether theorem. Finally, the three-dimensional diagrams of some obtained solutions are demonstrated utilizing Matlab for visualization, and some of the calculations were verified using computational packages Maple for symbolic computation.

Lie group analysis, solitary wave solutions and conservation laws of Schamel Burger’s equation

This paper presents a Lie group analysis of the Schamel Burger’s equation, notable for producing shock-type traveling waves in distinctive physical contexts. We determine the infinitesimal generators for this equation using the Lie group theory of differential equations. By applying Lie point symmetries, we establish commutation relations, the adjoint representation, and identify the optimal system of sub-algebras. Using elements from this optimal system, we perform symmetry reductions, resulting in various nonlinear ordinary differential equations (ODEs). Some of these reductions yield exact explicit solutions, while others necessitate the use of the new auxiliary equation method to obtain optical soliton solutions. We illustrate the dynamics of these soliton solutions graphically through both two and three-dimensional representations of wave structures. Additionally, we compute the conservation laws for the Schamel Burger’s equation by applying Ibragimov’s theorem, deriving conserved quantities corresponding to its point Lie symmetries. This analysis underscores our novel contribution, offering insights not previously explored in the literature.

Radially symmetric solutions of the ultra-relativistic Euler equations in several space dimensions

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure, the spatial part of the dimensionless four-velocity and the particle density. Radially symmetric solutions of these equations are studied in two and three space dimensions. Of particular interest in the solutions are the formation of shock waves and a pressure blow up. For the investigation of these phenomena we develop a one-dimensional scheme using radial symmetry and integral conservation laws. We compare the numerical results with solutions of multi-dimensional high-order numerical schemes for general initial data in two space dimensions. The presented test cases and results may serve as interesting benchmark tests for multi-dimensional solvers.

Description of step loads at high temperature creep of metallic materials

Abstract In metallic materials under high-temperature creep conditions, the damage evolution occurs. To describe it, Kachanov and Rabotnov proposed the concept of damage. In this paper, the damage parameter is defined as the relative change in material density, serving as an integral measure of damage. By incorporating the damage parameter and the mass conservation law, are we formulated interrelated kinetic equations for rate of creep deformation and the damage parameter. The case of variable loads is crucial both theoretically and practically, as most structural components typically operate under such conditions. Therefore, solutions of considered equations are derived for the case of two-stage loading. The received solutions and experimental results on the creep of titanium alloy under variable loads at 500°C were compared. There is a good agreement between the theoretical results and experimental data.

Deep finite volume method for partial differential equations

In this paper, we introduce the Deep Finite Volume Method (DFVM), an innovative deep learning framework tailored for solving high-order (order ≥2) partial differential equations (PDEs). Our approach centers on a novel loss function crafted from local conservation laws derived from the original PDE, distinguishing DFVM from traditional deep learning methods. By formulating DFVM in the weak form of the PDE rather than the strong form, we enhance accuracy, particularly beneficial for PDEs with less smooth solutions compared to strong-form-based methods like Physics-Informed Neural Networks (PINNs). A key technique of DFVM lies in its transformation of all second-order or higher derivatives of neural networks into first-order derivatives which can be computed directly using Automatic Differentiation (AD). This adaptation significantly reduces computational overhead, particularly advantageous for solving high-dimensional PDEs. Numerical experiments demonstrate that DFVM achieves equal or superior solution accuracy compared to existing deep learning methods such as PINN, Deep Ritz Method (DRM), and Weak Adversarial Networks (WAN), while drastically reducing computational costs. Notably, for PDEs with nonsmooth solutions, DFVM yields approximate solutions with relative errors up to two orders of magnitude lower than those obtained by PINN. The implementation of DFVM is available on GitHub at https://github.com/Sysuzqs/DFVM.

The combined effects of quantum and strong coupling on the nonlinear collective excitation in quantum dusty plasmas

The quantum hydrodynamic model for electrons and ions and the generalized hydrodynamic model for the strongly coupled dust particles are proposed in the strongly coupled quantum dusty plasma, where the combined quantum effects of quantum diffraction, quantum statistic pressure, as well as electron exchange and correlation effects are all considered in the quantum hydrodynamic model. The shear and bulk viscosity effects are included in the viscoelastic relaxation, which leads to the decay of the dust-ion-acoustic waves. The approximate time-dependent solitary solution is obtained by the momentum conservation law in the presence of viscosity.

On the study of an extended coupled KdV system: Analytical solutions and conservation laws

This paper aims to derive analytical solutions of an extended (2+1)-dimensional constant coefficients new coupled Korteweg–de Vries system. This will be achieved by implementing the classical symmetry method in conjunction with some simplest equation methods. The simplest equations that will be utilised includes among others the Bernoulli and Riccati equations. Furthermore, the conservation laws will be constructed through the multipliers approach, which subsequently reveals the conserved quantities. Moreover, a brief presentation of results obtained consisting of a variety of profile structures which include the kink type, bell and inverted bell shaped and singular wave solutions will be discussed.

Noether Symmetries of the Triple Degenerate DNLS Equations

In this paper, Lie symmetries and Noether symmetries along with the corresponding conservation laws are derived for weakly nonlinear dispersive magnetohydrodynamic wave equations, also known as the triple degenerate derivative nonlinear Schrödinger equations. The main goal of this study is to obtain Noether symmetries of the second-order Lagrangian density for these equations using the Noether symmetry approach with a gauge term. For this Lagrangian density, we compute the conserved densities and fluxes corresponding to the Noether symmetries with a gauge term, which differ from the conserved densities obtained using Lie symmetries in Webb et al. (J. Plasma Phys. 1995, 54, 201–244; J. Phys. A Math. Gen. 1996, 29, 5209–5240). Furthermore, we find some new Lie symmetries of the dispersive triple degenerate derivative nonlinear Schrödinger equations for non-vanishing integration functions Ki(t) (i=1,2,3).

OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws

Suppressing spurious oscillations is crucial for designing reliable high-order numerical schemes for hyperbolic conservation laws, yet it has been a challenge actively investigated over the past several decades. This paper proposes a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique (see J. Lu, Y. Liu, and C. W. Shu [SIAM J. Numer. Anal. 59 (2021), pp. 1299–1324]). The OEDG method incorporates an oscillation-eliminating (OE) procedure after each Runge–Kutta stage, and it is devised by alternately evolving the conventional semidiscrete discontinuous Galerkin (DG) scheme and a damping equation. A novel damping operator is carefully designed to possess both scale-invariant and evolution-invariant properties. We rigorously prove the optimal error estimates of the fully discrete OEDG method for smooth solutions of linear scalar conservation laws. This might be the first generic fully discrete error estimate for nonlinear DG schemes with an automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems spanning various scales and wave speeds, without necessitating problem-specific parameters for all the tested cases. It also obviates the need for characteristic decomposition in hyperbolic systems. Furthermore, it retains the key properties of the conventional DG method, such as local conservation, optimal convergence rates, and superconvergence. Moreover, the OEDG method maintains stability under the normal Courant–Friedrichs–Lewy (CFL) condition, even in the presence of strong shocks associated with highly stiff damping terms. The OE procedure is nonintrusive, facilitating seamless integration into existing DG codes as an independent module. Its implementation is straightforward and efficient, involving only simple multiplications of modal coefficients by scalars. The OEDG approach provides new insights into the damping mechanism for oscillation control. It reveals the role of the damping operator as a modal filter, establishing close relations between the damping technique and spectral viscosity techniques. Extensive numerical results validate the theoretical analysis and confirm the effectiveness and advantages of the OEDG method.

Pseudo grid-based physics-informed convolutional-recurrent network solving the integrable nonlinear lattice equations

Traditional discrete learning methods involve discretizing continuous equations using difference schemes, necessitating considerations of stability and convergence. Integrable nonlinear lattice equations possess a profound mathematical structure that enables them to revert to continuous integrable equations in the continuous limit, particularly retaining integrable properties such as conservation laws, Hamiltonian structure, and multiple soliton solutions. The pseudo grid-based physics-informed convolutional-recurrent network (PG-PhyCRNet) is proposed to investigate the localized wave solutions of integrable lattice equations, which significantly enhances the model’s extrapolation capability to lattice points beyond the temporal domain. We conduct a comparative analysis of PG-PhyCRNet with and without pseudo grid by investigating the multi-soliton solutions and rational solitons of the Toda lattice and self-dual network equation. The results indicate that the PG-PhyCRNet excels in capturing long-term evolution and enhances the model’s extrapolation capability for solitons, particularly those with steep waveforms and high wave speeds. Finally, the robustness of the PG-PhyCRNet method and its effect on the prediction of solutions in different scenarios are confirmed through repeated experiments involving pseudo grid partitioning.

2-Site versus 3-site models of ATP hydrolysis by F1-ATPase: definitive mathematical proof using combinatorics and conservation equations.

The F1-ATPase enzyme is the smallest-known molecular motor that rotates in 120° steps, driven by the hydrolysis of ATP. It is a multi-subunit enzyme that contains three catalytic sites. A central question is how the elementary chemical reactions that occur in the three sites are coupled to mechanical rotation. Various models and coupling schemes have been formulated in an attempt to answer this question. They can be classified as 2-site (bi-site) models, exemplified by Boyer's binding change mechanism first proposed 50years ago, and 3-site (tri-site) models such as Nath's torsional mechanism, first postulated 25years ago and embellished 1year back. Experimental data collated using diverse approaches have conclusively shown that steady-state ATP hydrolysis by F1-ATPase occurs in tri-site mode. Hence older models have been continually modified to make them conform to the new facts. Here, we have developed a pure mathematical approach based on combinatorics and conservation laws to test if proposed models are 2-site or 3-site. Based on this novel combinatorial approach, we have proved that older and modified models are effectively bi‒site models in that catalysis and rotation in F1-ATPase occurs in these models with only two catalytic sites occupied by bound nucleotide. Hence these models contradict consensus experimental data. The recent 2023 model of ATP hydrolysis by F1-ATPase has been proved to be a true tri-site model based on our novel mathematical approach. Such pure mathematical proofs constitute an important step forward for ATP mechanism. However, in what must be considered an aspect with great scientific potential, the power of such mathematical proofs has not been fully exploited to solve molecular biological problems, in our opinion. We believe that the creative application of pure mathematical proofs (for another example see Nath in Theory Biosci 141:249-260, 2022) can help resolve with finality various longstanding molecular-level issues that arise as a matter of course in the analysis of fundamental biological problems. Such issues have proved extraordinarily difficult to resolve by standard experimental, theoretical, or computational approaches.

Inverse Lax-Wendroff Boundary Treatment for Solving Conservation Laws with Finite Volume Methods

Inverse Lax-Wendroff Boundary Treatment for Solving Conservation Laws with Finite Volume Methods

Mechanical properties and energy evolution mechanism of wheat grain under uniaxial compression

To determine the mechanical properties and energy evolution characteristics of wheat kernels during compression, the mechanical properties and energy evolution of wheat kernels with high-gluten, medium-gluten, and low-gluten were analyzed in uniaxial compression test. Based on the theory of Hertz's formula, the stress-strain laws of the three varieties of wheat kernels in the process of compression damage were investigated; Based on the law of energy conservation, the energy evolution characteristics of the three varieties of wheat kernels under the same compression conditions were analyzed. The results showed that the stress-strain process of wheat kernels with high-gluten, medium-gluten, and low-gluten under compressive stress followed a similar trend overall, with the corresponding minimum failure strengths at the peak being 51.79 MPa, 50.07 MPa, and 46.21 MPa, respectively. The energy evolution patterns of three varieties of wheat kernels under uniaxial compressive loading were approximately the same, indicating that the elastic energy was mainly accumulated before the peak and reached the maximum at the peak. In contrast, the proportion of dissipated energy increases after the peak and rapidly exceeds the elastic energy. In this paper, the damage characteristics of wheat kernels are described from the energy point of view, which is of great significance for the new conceptualization of the evolution of wheat grain damage.

Numerical Simulation of Kink Collisions, Analytical Solutions and Conservation Laws of the Potential Korteweg–de Vries Equation

Numerical Simulation of Kink Collisions, Analytical Solutions and Conservation Laws of the Potential Korteweg–de Vries Equation

Cylinder-piston application for surge prevention and energy efficiency in an industrial compression unit

Compressions are prevalent in industrial applications and are notable for their substantial energy consumption. Therefore, the simulation and analysis of the compression process are essential for maintenance and energy conservation efforts. These systems are prone to potentially unstable surge conditions, necessitating the use of traditional anti-surge valves that result in considerable energy losses. Ensuring the near-optimal operation of these systems is critical to minimizing energy consumption.In this article, a conceptual framework for a cylinder-piston mechanism is delineated, intended for design and operation as an active surge control system. Additionally, a modular quasi-one-dimensional model is articulated for the transient simulation of an industrial compression system, which integrates models for both the anti-surge and active control systems. The manuscript presents a novel design, featured by a cylinder-piston system integrated with a robust controller, posited as a potential alternative to traditional anti-surge systems. The effectiveness of this design in expanding the operational envelope of the compression system and surge prevention is rigorously examined. Moreover, a thermodynamic model, grounded in the fundamental laws of mass, momentum, and energy conservation, is applied to each component of the system. Furthermore, the manuscript explores the benefits of the innovative design in achieving a marked decrease in energy wastage. Simulation results from a test scenario reveals that the implementation of the cylinder-piston design, as opposed to the conventional anti-surge system, can diminish energy losses and associated pollutant emissions by approximately 33 percent.

How general is the Jensen–Varadhan large deviation functional for 1D conservation laws?

Starting from a microscopic particle model whose hydrodynamic limit under hyperbolic space-time scaling is a 1D conservation law, we derive the large deviation rate function encoding the probability to observe a density profile which is a non entropic shock, and compare this large deviation rate function with the classical Jensen-Varadhan functional, valid for the totally asymmetric exclusion process and the weakly asymmetric exclusion process in the strong asymmetry limit. We find that these two functionals have no reason to coincide, and in this sense Jensen-Varadhan functional is not universal. However, they do coincide in a small Mach number limit, suggesting that universality is restored in this regime. We then compute the leading order correction to the Jensen-Varadhan functional.

Reduced floating-point precision in regional climate simulations: an ensemble-based statistical verification

Abstract. The use of single precision in floating-point representation has become increasingly common in operational weather prediction. Meanwhile, climate simulations are still typically run in double precision. The reasons for this are likely manifold and range from concerns about compliance and conservation laws to the unknown effect of single precision on slow processes or simply the less frequent opportunity and higher computational costs of validation. Using an ensemble-based statistical methodology, Zeman and Schär (2022) could detect differences between double- and single-precision simulations from the regional weather and climate model COSMO. However, these differences are minimal and often only detectable during the first few hours or days of the simulation. To evaluate whether these differences are relevant for regional climate simulations, we have conducted 10-year-long ensemble simulations over the European domain of the Coordinated Regional Climate Downscaling Experiment (EURO-CORDEX) in single and double precision with 100 ensemble members. By applying the statistical testing at a grid-cell level for 47 output variables every 12 or 24 h, we only detected a marginally increased rejection rate for the single-precision climate simulations compared to the double-precision reference based on the differences in distribution for all tested variables. This increase in the rejection rate is much smaller than that arising from minor variations of the horizontal diffusion coefficient in the model. Therefore, we deem it negligible as it is masked by model uncertainty. To our knowledge, this study represents the most comprehensive analysis so far on the effects of reduced precision in a climate simulation for a realistic setting, namely with a fully fledged regional climate model in a configuration that has already been used for climate change impact and adaptation studies. The ensemble-based verification of model output at a grid-cell level and high temporal resolution is very sensitive and suitable for verifying climate models. Furthermore, the verification methodology is model-agnostic, meaning it can be applied to any model. Our findings encourage exploiting the reduction of computational costs (∼30 % for COSMO) obtained from reduced precision for regional climate simulations.