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The compact finite difference method is a powerful tool for discretizing conservation laws, owing to its inherent flexibility in developing high-resolution and highly stable schemes. In this paper, we propose a framework for the design of genuine globally conservative compact finite difference schemes, which addresses a critical requirement in conservation laws. Within our framework, we rigorously establish that the discrete conservation law maintains strict conservation for flux functions in polynomial spaces with optimal algebraic order, i.e., the discrete scheme achieves an optimal algebraic precision.Our work advances the existing conservative compact finite difference schemes, which rely on approaches to maintaining global conservation that are fundamentally consistent with the method proposed by Lele [Lele, J. Comput. Phys., 1992]. As an application, we propose an algorithm for designing globally conservative fourth-order schemes, aimed at optimizing resolution and asymptotic stability. Three schemes are generated using the algorithm, with their excellent performance across multiple aspects validated through numerical experiments.

While conservation laws in gradient flow training dynamics are well understood for (mostly shallow) ReLU and linear networks, their study remains largely unexplored for more practical architectures. This paper bridges this gap by deriving and analyzing conservation laws for modern architectures, with a focus on convolutional ResNets and Transformer networks. For this, we first show that basic building blocks such as ReLU (or linear) shallow networks, with or without convolution, have easily expressed conservation laws, and no more than the known ones. In the case of a single attention layer, we also completely describe all conservation laws, and we show that residual blocks have the same conservation laws as the same block without a skip connection. We then introduce the notion of conservation laws that depend only on a subset of parameters (corresponding e.g. to a pair of consecutive layers, to a residual block, or to an attention layer). We demonstrate that the characterization of such laws can be reduced to the analysis of the corresponding building block in isolation. Finally, we examine how these newly discovered conservation principles, initially established in the continuous

In a recent paper, PNAS, 118, e1921529118 (2021), it was argued that while the standard definition of conservation laws in quantum mechanics, which is of a statistical character, is perfectly valid, it misses essential features of nature and it can and must be revisited to address the issue of conservation/non-conservation in individual cases. Specifically, in the above paper an experiment was presented in which it can be proven that in some individual cases energy is not conserved, despite being conserved statistically. It was felt however that this is worrisome, and that something must be wrong if there are individual instances in which conservation doesn't hold, even though this is not required by the standard conservation law. Here we revisit that experiment and show that although its results are correct, there is a way to circumvent them and ensure individual case conservation in that situation. The solution is however quite unusual, challenging one of the basic assumptions of quantum mechanics, namely that any quantum state can be prepared, and it involves a time-holistic, double non-conservation effect. Our results bring new light on the role of the preparation stage of the

A complete and explicit classification of all independent local conservation laws of Maxwell's equations in four dimensional Minkowski space is given. Besides the elementary linear conservation laws, and the well-known quadratic conservation laws associated to the conserved stress-energy and zilch tensors, there are also chiral quadratic conservation laws which are associated to a new conserved tensor. The chiral conservation laws possess odd parity under the electric-magnetic duality transformation of Maxwell's equations, in contrast to the even parity of the stress-energy and zilch conservation laws. The main result of the classification establishes that every local conservation law of Maxwell's equations is equivalent to a linear combination of the elementary conservation laws, the stress-energy and zilch conservation laws, the chiral conservation laws, and their higher order extensions obtained by replacing the electromagnetic field tensor by its repeated Lie derivatives with respect to the conformal Killing vectors on Minkowski space. The classification is based on spinorial methods and provides a direct, unified characterization of the conservation laws in terms of Killing sp

For systems of partial differential equations in three spatial dimensions, dynamical conservation laws holding on volumes, surfaces, and curves, as well as topological conservation laws holding on surfaces and curves, are studied in a unified framework. Both global and local formulations of these different conservation laws are discussed, including the forms of global constants of motion. The main results consist of providing an explicit characterization for when two conservation laws are locally or globally equivalent, and for when a conservation law is locally or globally trivial, as well as deriving relationships among the different types of conservation laws. In particular, the notion of a ``trivial'' conservation law is clarified for all of the types of conservation laws. Moreover, as further new results, conditions under which a trivial local conservation law on a domain can yield a non-trivial global conservation law on the domain boundary are determined and shown to be related to differential identities that hold for PDE systems containing both evolution equations and spatial constraint equations. Numerous physical examples from fluid flow, gas dynamics, electromagnetism, a

A high order time stepping applied to spatial discretizations provided by the method of lines for hyperbolic conservations laws is presented. This procedure is related to the one proposed in Qiu and Shu (SIAM J Sci Comput 24(6):2185-2198, 2003) for numerically solving hyperbolic conservation laws. Both methods are based on the conversion of time derivatives to spatial derivatives through a Lax-Wendroff-type procedure, also known as Cauchy-Kovalevskaya process. The original approach in Qiu and Shu (2003) uses the exact expressions of the fluxes and their derivatives whereas the new procedure computes suitable finite difference approximations of them ensuring arbitrarily high order accuracy both in space and time as the original technique does, with a much simpler implementation and generically better performance, since only flux evaluations are required and no symbolic computations of flux derivatives are needed.

Potential vorticity (PV) is one of the most important quantities in atmospheric science. The PV of each fluid parcel is known to be conserved in the case of a dry atmosphere. However, a parcel's PV is not conserved if clouds or phase changes of water occur. Recently, PV conservation laws were derived for a cloudy atmosphere, where each parcel's PV is not conserved but parcel-integrated PV is conserved, for integrals over certain volumes that move with the flow. Hence a variety of different statements are now possible for moist PV conservation and non-conservation, and in comparison to the case of a dry atmosphere, the situation for moist PV is more complex. Here, in light of this complexity, several different definitions of moist PV are compared for a cloudy atmosphere. Numerical simulations are shown for a rising thermal, both before and after the formation of a cloud. These simulations include the first computational illustration of the parcel-integrated, moist PV conservation laws. The comparisons, both theoretical and numerical, serve to clarify and highlight the different statements of conservation and non-conservation that arise for different definitions of moist PV.

For polynomial ODE models, we introduce and discuss the concepts of exact and approximate conservation laws, which are the first integrals of the full and truncated sets of ODEs. For fast-slow systems, truncated ODEs describe the fast dynamics. We define compatibility classes as subsets of the state space, obtained by equating the conservation laws to constants. A set of conservation laws is complete when the corresponding compatibility classes contain a finite number of steady states. Complete sets of conservation laws can be used for model order reduction and for studying the multistationarity of the model. We provide algorithmic methods for computing linear, monomial, and polynomial conservation laws of polynomial ODE models and for testing their completeness. The resulting conservation laws and their completeness are either independent or dependent on the parameters. In the latter case, we provide parametric case distinctions. In particular, we propose a new method to compute polynomial conservation laws by comprehensive Gröbner systems and syzygies. Keywords: First integrals, chemical reaction networks, polynomial conservation laws, syzygies, comprehensive Gröbner systems.

Market-based instruments such as payments, auctions or tradable permits have been proposed as flexible and cost-effective instruments for biodiversity conservation on private lands. Trading the service of conservation requires one to define a metric that determines the extent to which a conserved site adds to the regional conservation objective. Yet, while markets for conservation are widely discussed and increasingly applied, little research has been conducted on explicitly accounting for spatial ecological processes in the trading. In this paper, we use a coupled ecological economic simulation model to examine how spatial connectivity may be considered in the financial incentives created by a market-based conservation scheme. Land use decisions, driven by changing conservation costs and the conservation market, are simulated by an agent-based model of land users. On top of that, a metapopulation model evaluates the conservational success of the market. We find that optimal spatial incentives for agents correlate with species characteristics such as the dispersal distance, but they also depend on the spatio-temporal distribution of conservation costs. We conclude that a combined a

Conservation of current and conservation of charge are nearly the same thing: when enough is known about charge movement, conservation of current can be derived from conservation of charge, in ideal dielectrics, for example. Conservation of current is enforced implicitly in ideal dielectrics by theories that conserve charge. But charge movement in real materials like semiconductors or ionic solutions is never ideal. We present an apparently universal derivation of conservation of current and advocate using that conservation law explicitly as a distinct part of theories and calculations of charge movement in complex fluids and environments. Classical models using ordinary differential equations rarely satisfy conservation of current, including the chemical kinetic models implementing the law of mass action and Markov models. These models must be amended if they are to conserve current. Strict enforcement of conservation of current is likely to aid numerical analysis by preventing artifactual accumulation of charge.

In this white paper, we synthesize key points made during presentations and discussions from the AI-Assisted Decision Making for Conservation workshop, hosted by the Center for Research on Computation and Society at Harvard University on October 20-21, 2022. We identify key open research questions in resource allocation, planning, and interventions for biodiversity conservation, highlighting conservation challenges that not only require AI solutions, but also require novel methodological advances. In addition to providing a summary of the workshop talks and discussions, we hope this document serves as a call-to-action to orient the expansion of algorithmic decision-making approaches to prioritize real-world conservation challenges, through collaborative efforts of ecologists, conservation decision-makers, and AI researchers.

The interrelationship between energy and probability conservation is explored from the point of view of statistical physics and non-relativistic quantum mechanics. The simultaneous validity of the law of conservation of energy and the continuity equation (probability conservation) breaks for an interacting dynamical system. A separate and independent description of a physical system can be obtained by requiring that the law of conservation of probability is at the heart of the derivation of the ''equations of motion''. In effect, The Schrodinger equation can be viewed as an appropriate factorization of the continuity equation instead of an energy conservation relation per se.

Despite the large number of publications on symmetry analysis of the geopotential forecast equation, its group foliations laws have not been considered previously. The present publication aims to address this shortcoming. First, group foliations are constructed for the equation, and based on them, invariant solutions are derived, some of which generalize previously known exact solutions. There is also a discussion of the pros and cons of the group foliation approach. In addition, the rest of the paper is dedicated to a comprehensive discussion of conservation laws. All possible second-order conservation laws of the geopotential forecast equation are obtained through direct calculations, and a number of higher-order conservation laws are derived using the known symmetries of the equation.

This paper presents recent work on connections between symmetries and conservation laws. After reviewing Noether's theorem and its limitations, we present the Direct Construction Method to show how to find directly the conservation laws for any given system of differential equations. This method yields the multipliers for conservation laws as well as an integral formula for corresponding conserved densities. The action of a symmetry (discrete or continuous) on a conservation law yields conservation laws. Conservation laws yield non-locally related systems that, in turn, can yield nonlocal symmetries and in addition be useful for the application of other mathematical methods. From its admitted symmetries or multipliers for conservation laws, one can determine whether or not a given system of differential equations can be linearized by an invertible transformation.

The Lie-point symmetry method is used to find some closed-form solutions for a constitutive equation modeling stress in elastic materials. The partial differential equation (PDE), which involves a power law with arbitrary exponent n, was investigated by Mason and his collaborators (Magan et al., Wave Motion, 77, 156-185, 2018). The Lie algebra for the model is five-dimensional for the shearing exponent n > 0, and it includes translations in time, space, and displacement, as well as time-dependent changes in displacement and a scaling symmetry. Applying Lie's symmetry method, we compute the optimal system of one-dimensional subalgebras. Using the subalgebras, several reductions and closed-form solutions for the model are obtained both for general exponent n and special case n = 1. Furthermore, it is shown that for general n > 0 the model has interesting conservation laws which are computed with symbolic software using the scaling symmetry of the given PDE.

In this paper, a modified version of conservative Physics-informed Neural Networks (cPINN for short) is provided to construct the weak solutions of Riemann problem for the hyperbolic scalar conservation laws in non-conservative form. To demonstrate the results, we use the model of generalized Buckley-Leverett equation (GBL equation for short) with discontinuous porosity in porous media. By inventing a new unknown, the GBL equation is transformed into a two-by-two resonant hyperbolic conservation laws in conservative form. The modified method of cPINN is invented to overcome the difficulties due to the discontinuity of the porosity and the appearance of the critical states (near vacuum) in the Riemann data. We experiment with our idea by using a deep learning algorithm to solve the GBL equation in both conservative and non-conservative forms, as well as the cases of critical and non-critical states. This method provides a combination of two different neural networks and corresponding loss functions, one is for the two-by-two resonant hyperbolic system, and the other is for the scalar conservation law with a discontinuous perturbation term in the non-convex flux. The technique of re-

The explicit formulation of the general inverse problem on conservation laws is presented for the first time. In this problem one aims to derive the general form of systems of differential equations that admit a prescribed set of conservation laws. The particular cases of the inverse problem on first integrals of ordinary differential equations and on conservation laws for evolution equations are studied. We also solve the inverse problem on conservation laws for differential equations admitting an infinite dimensional space of zeroth-order conservation-law characteristics. This particular case is further studied in the context of conservative first-order parameterization schemes for the two-dimensional incompressible Euler equations. We exhaustively classify conservative first-order parameterization schemes for the eddy-vorticity flux that lead to a class of closed, averaged Euler equations possessing generalized circulation, generalized momentum and energy conservation.

For each substance-like quantity, a theorem about its conservation or non-conservation can be formulated. For the electric charge e.g. it reads: Electric charge can neither be created nor destroyed. Such a statement is short and easy to understand. For some quantities, however, the proposition about conservation or non-conservation is usually formulated in an unnecessarily complicated way. Sometimes the formulation is not generally valid; in other cases only a consequence of the conservation or non-conservation is pronounced. A clear and unified formulation could improve the comprehensibility and simplify teaching.

The paper discusses impact of the velocity curl on some conservation laws in the gravitational field and electromagnetic field, by means of the characteristics of quaternions. When the velocity curl can not be neglected, it will cause the predictions to departure slightly from the conservation laws, which include mass continuity equation, charge continuity equation, and conservation of spin, etc. And the scalar potential of gravitational field has an effect on the speed of light, the conservation of mass, and conservation of charge, etc. The results explain how the velocity curl influences some conservation laws in the gravitational field and electromagnetic field.