Conserved quantities on multisymplectic manifolds extend symplectic geometry results.
problem Defining and analyzing conserved quantities on multisymplectic manifolds.
method Defining globally conserved quantities as Lie derivatives of exact forms, extending classical results in symplectic geometry.
result A family of globally conserved quantities can be obtained under certain symmetries and homotopy co-momentum maps.
Following an approach of the second author for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuit…
ECD algorithm speeds up non-convex optimization, offering quantum and stochastic enhancements.
problem Non-convex optimization challenges in machine learning.
method Energy Conserving Descent (ECD) algorithm, stochastic ECD dynamics (sECD), quantum ECD Hamiltonian (qECD).
result ECD and its quantum version achieve exponential speedup over gradient descent.
Study maps research streams in biodiversity finance, identifies key areas.
problem Biodiversity loss and need for finance to reverse trends.
method Quantitative bibliometric analysis of 189,456 references.
result Identifies eight primary research streams in biodiversity finance.
Global solution found for biharmonic wave maps into spheres.
problem Solving biharmonic wave maps into spherical targets.
method Reformulated as a conservation law, solved with Ginzburg-Landau approximation.
result Global weak solution constructed in the energy space.
In the classical Lagrangian approach to conservation laws of gauge-natural field theories a suitable (vector) density is known to generate the so--called {\em conserved Noether currents}. It turns out that along any section of the relevant gauge--natural bundle this density is the divergence of a skew--symmetric (tenso…
GoT-WAVE improves temporal network alignment by 25% accuracy and 64% speed.
problem Finding conserved network regions in temporal networks.
method Using graphlet-orbit transitions (GoTs) as a dynamic node similarity measure within DynaWAVE.
result GoT-WAVE outperforms DynaWAVE in accuracy and speed on synthetic networks.
We derive both {\em local} and {\em global} generalized {\em Bianchi identities} for classical Lagrangian field theories on gauge-natural bundles. We show that globally defined generalized Bianchi identities can be found without the {\em a priori} introduction of a connection. The proof is based on a {\em global} decom…
Unified neural network framework for context-aware Gaussian overbounds in uncertainty propagation.
problem Uncertainty quantification in safety-critical settings requires conservative bounds, but existing methods often fail to compose and are overly conservative.
method Proposes a learning framework that trains neural networks to produce context-aware Gaussian overbounds with provable conservatism.
result The method yields tighter bounds while maintaining conservatism on the enforced grid and in experiments.
Gradient descent dynamics studied for DEQs in linear and single-index models.
problem Understanding gradient descent dynamics for DEQs.
method Rigorously studied gradient descent dynamics for DEQs in linear and single-index models.
result Gradient descent converges to a global minimizer for linear DEQs and single-index models.
Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in very general setting. The cohomology of the variational bicomplex on an arbitrary graded manifold and the iterated cohomology of a generic nilpotent contact supersymmetry are computed. In particular, the first variational fo…
The paper proves curvature pinching theorems and vanishing results for Riemannian manifolds.
problem Curvature pinching and vanishing theorems for Riemannian manifolds.
method Monotonicity formulas for vector bundle-valued p-forms, conservation law, and Li-Tam's theorem for harmonic functions.
result Under curvature pinching conditions, Riemannian manifolds have only one end and satisfy vanishing theorems for harmonic forms.
Paper bridges matching rules and height functions in aperiodic tilings.
problem Relationship between matching rules and height functions in aperiodic tilings.
method Cochain-first framework to establish equivalence between matching rules, Ammann bar continuity, cycle closure of 1-cochains, and height-function existence.
result Unified framework for aperiodic tilings including Penrose and canonical projection tilings.
Improved weather forecasting using deep CNN on cubed-sphere grid.
problem Global weather prediction accuracy and speed.
method Deep convolutional neural network (CNN) on cubed-sphere grid, offline mapping, loss minimization.
result Significantly improved weather forecasts, indefinitely stable, realistic patterns at long lead times.
Theoretical analysis confirms non-conservative algorithms can converge to optimal policies.
problem Theoretical guarantees for non-conservative reinforcement learning algorithms.
method Theoretical analysis of Peng's Q(λ) algorithm. result Peng's Q(λ) converges to an optimal policy under certain conditions. Study finds conservation laws for a specific class of parabolic equations.
problem Existence and structure of conservation laws for evolutionary scalar second-order differential equations.
method Calculation of linearized characteristic cohomology to find conservation laws, showing dependence on second derivatives.
result Only Monge-Ampère type equations have non-trivial conservation laws.
An analytic extension of the Reissner-Nordstrom solution at and beyond the singularity is presented. The extension is obtained by using new coordinates in which the metric becomes degenerate at r=0. The metric is still singular in the new coordinates, but its components become finite and smooth. Using this extension …
Geometric technique determines exactness of SDP robustness certificate.
problem Certifying robustness of neural networks to adversarial examples.
method Geometric projection onto hyperbola, SDP relaxation of ReLU activation.
result SDP certificate is exact for a single hidden layer under mild assumptions.
Study finds conserved quantities for two types of curves on conformal sphere.
problem Identifying conserved quantities for specific types of curves on a conformal sphere.
method Used parallel tractor and Lagrangian formalism to compute conserved quantities.
result Found relation between conserved quantities of two curve types.
Using the symmetry group theory of second order PDEs, one finds the symmetry group associated to Tzitzeica surfaces partial differential equation. One studies the inverse problem and one shows that the Tzitzeica surfaces PDE is an Euler-Lagrange equation. One determines the variational symmetry group of the associated …
Survey on conservation laws for geometric PDEs.
problem Modeling polyharmonic maps.
method Conservation law approach.
result Overview of conservation laws in geometric PDEs.
The article discusses conservation laws for polyharmonic maps and their applications.
problem Understanding conservation laws for polyharmonic maps.
method Recalling the stress-energy tensor and showing conservation laws with Killing vector fields.
result Conservation laws for polyharmonic maps and their applications.
Algorithm reconstructs conserved networks from flow data.
problem Network reconstruction from flow data.
method Polynomial time algorithm exploiting graph theoretic properties and learning techniques.
result Exact network reconstruction possible for arborescence networks.
Paper presents a reduction-based framework for conservative bandits and RL with improved lower and upper bounds.
problem Conservative bandits and reinforcement learning problems.
method Reduction technique to calculate necessary and sufficient budget from baseline policy.
result Improved lower and upper bounds for various conservative settings.
Researchers found a geometric duality for systems of conservation laws.
problem Systems of conservation laws and their additional conservation laws.
method Assigning ruled surfaces in projective space and defining dual systems.
result Hamiltonian systems are autodual, and 3-component nondiagonalizable systems are dual to systems with constant characteristic speeds.
Study finds conservation laws for maps on spheres.
problem Conservation laws for maps on spheres.
method Derives conservation laws for Dirac-harmonic maps and their extensions.
result Conservation laws for maps on spheres.
Non-trivial conservation law found for a specific system.
problem Conservation law for a specific system with a vanishing characteristic.
method Analyzing overdetermined system with given characteristics.
result Non-trivial conservation law despite vanishing characteristic.
Mathematical model describes how red blood cells return to equilibrium.
problem How red blood cells regain equilibrium after deformation.
method Gradient flow of the Canham-Helfrich functional, proving global existence and convergence for spheres and axisymmetric tori.
result Global existence and convergence of smooth solutions for spheres and axisymmetric tori under specific energy conditions.
This work connects symmetries and conserved quantities in machine learning.
problem Improving machine learning models by learning conserved quantities.
method Using Noether's theorem, learn symmetries and conserved quantities directly from data.
result Correctly identifies conserved quantities and improves model performance.
Proposes a conservative exploration method for RL agents.
problem Guaranteeing performance of exploratory policies in RL.
method Importance sampling for off-policy policy evaluation.
result Derives a regret bound ensuring no conservative constraint violation.
The conservation laws of the third order quasilinear scalar evolution equations are considered via differential system and characteristic cohomology. We find a subspace of 2 forms in the infinite prolonged space in which every conservation law has a unique representative. The structure of this subspace naturally gives …
Conservation law for weakly harmonic mappings in high dimensions.
problem Conservation law for harmonic mappings in supercritical dimensions.
method Partial extension of Rivière's conservation law with Lorentz integrability condition.
result Conservation law for weakly harmonic mappings in supercritical dimensions.
We study higher-order conservation laws of the non-linearizable elliptic Poisson equation ∂z∂zˉ∂2u=−f(u) as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for diffe…
New conservation laws found for polyharmonic maps in critical dimension.
problem Existence of conservation laws for polyharmonic maps in critical dimension.
method Small perturbation of Uhlenbeck's gauge fixing matrix.
result Existence of conservation laws for elliptic systems of even order in critical dimension.
New variational principle found for PDEs with symmetries and conservation laws.
problem Finding variational principles for PDEs with symmetries and conservation laws.
method Proving existence of a variational principle for PDEs with symmetries and conservation laws.
result A differential equation with sufficient symmetries and conservation laws leads to a variational functional.
MC-LSTM extends LSTM to conserve mass in neural networks.
problem Conservation laws in real-world systems.
method Extending LSTM's inductive bias to conserve mass.
result MC-LSTM sets new state-of-the-art for predicting peak flows.
The paper studies symmetries and conservation laws of non-diagonalisable hydrodynamic systems.
problem Integrating non-diagonalisable hydrodynamic systems of partial differential equations.
method Analysis of gl-regular Nijenhuis operators, splitting Theorem for symmetries and conservation laws, relationship between symmetries and conservation laws.
result The system of partial differential equations is integrable in quadratures.
Unified physics field theories through a general conservation law.
problem Unified physics field theories.
method Introduced general field as a formal sum of differential forms, defined conservation law using action principle.
result Physics field theories become instances of the general conservation law.
We present a connection between the Killing fields that arise in the loop-group approach to integrable systems and conservation laws viewed as elements of the characteristic cohomology. We use the connection to generate the complete set of conservation laws (as elements of the characteristic cohomology) for the Tzitzei…
We propose a Markov chain model for credit rating changes. We do not use any distributional assumptions on the asset values of the rated companies but directly model the rating transitions process. The parameters of the model are estimated by a maximum likelihood approach using historical rating transitions and heurist…
A new algorithm balances exploration and exploitation in online decision-making.
problem Balancing exploration and exploitation in online decision-making.
method Proposed C4-UCB algorithm incorporating conservative mechanism. result Proved n-step upper regret bound for two situations.
We obtain necessary and sufficient conditions for the existence of "conservation laws" on null hypersurfaces for the wave equation on general four-dimensional Lorentzian manifolds. Examples of null hypersurfaces exhibiting such conservation laws include the standard null cones of Minkowski spacetime and the degenerate …
New neural network enforces mass conservation for better ice flow predictions.
problem Reliably project future sea level rise by improving ice sheet model inputs.
method Proposes divergence-free neural networks (dfNNs) enforcing local mass conservation.
result dfNNs yield more reliable ice flux estimates compared to other models.
This paper contains the second part of a two-part series on the stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial da…
The study finds resonance points in polarised curves with polynomial conserved quantities.
problem Finding resonance points in polarised curves with polynomial conserved quantities.
method Using the non-orthogonality assumption on the conserved quantity, the study deduces the existence of resonance points.
result Every finite type polarised curve in the conformal 2-sphere with a polynomial conserved quantity admits a resonance point.
There is a well-known example of integrable conservative system on S2, the case of Kovalevskaya in the dynamics of a rigid body, possessing an integral of fourth degree in momenta. Goryachev proposed a one-parameter family of examples of conservative systems on S2 possessing an integral of fourth degree in moment…
Deep learning improves climate models by capturing sub-grid processes efficiently.
problem Uncertainty in climate models due to sub-grid processes, especially clouds.
method Trained a deep neural network to represent atmospheric sub-grid processes in a climate model.
result The neural network accurately reproduces climate and variability features from a cloud-resolving model.
Data symmetries in neural networks can generate conserved quantities.
problem Conservation laws in neural networks
method Using tensorizable networks
result Data augmentation can induce conserved quantities