For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well known singularities of generic vector fields on the plane and the singularit…
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48 results for “implicit equations”
In this paper, we make a generalization of Routh's reduction method for Lagrangian systems with symmetry to the case where not any regularity condition is imposed on the Lagrangian. First, we show how implicit Lagrange-Routh equations can be obtained from the Hamilton-Pontryagin principle, by making use of an anholonom…
Abstract: Study of surface transitions and IDE inflections via contact geometry.
problem Understanding transitions on surfaces and implicit differential equations.
method Contact geometry and Legendrian properties of projections.
result List of unavoidable local phenomena on surfaces and IDE solutions.
A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie groupoid may be described in terms of Lagrangian implicit difference equations …
SINDy-PI robustly identifies implicit dynamics from noisy data.
problem Accurately modeling nonlinear dynamics from noisy data.
method Parallel, implicit SINDy algorithm with multiple optimization algorithms and model selection.
result Significantly more noise robust than previous SINDy approaches.
Solves inverse problem for Maxwell equations using vector fields.
problem Inverse problem for Maxwell equations in vacuum.
method Abstract theory of implicit differential equations over pre-symplectic manifolds.
result Provides solution for Maxwell equations using vector fields.
Gradient descent converges to a global minimum in nonlinear ReLU implicit networks with linear width.
problem Understanding convergence of gradient methods in nonlinear, infinitely deep ReLU networks.
method Introduced a scaling constant to ensure well-posedness of the equilibrium equation, proving convergence to a global minimum for linear width networks.
result Gradient descent converges to a global minimum at a linear rate for nonlinear ReLU implicit networks with linear width.
This paper proves SGD converges to global minimum for over-parameterized ReLU networks.
problem Theoretical understanding of implicit neural networks is limited.
method Gradient flow analysis of ReLU activated implicit neural networks.
result Randomly initialized gradient descent converges to global minimum at a linear rate for square loss function in over-parameterized ReLU networks.
Unified framework for implicit generative models with theoretical guarantees.
problem Learning implicit generative models with theoretical guarantees.
method Integrating optimal transport, numerical ODE, density-ratio estimation, and deep neural networks.
result Unified framework with theoretical guarantees for implicit generative learning.
A Logic Based Approach to Finding Real Singularities of Implicit Ordinary Differential Equationsmath.LO
Logic approach finds real singularities in differential equations.
problem Finding geometric singularities of implicit ODEs over the reals.
method Vessiot theory, parametric Gaussian elimination, heuristic simplification, real quantifier elimination.
result Effective computation of geometric singularities using logic methods.
The study identifies unique Delaunay surfaces with constant mean curvature.
problem Characterizing surfaces with constant mean curvature.
method Analyzing surfaces defined by specific implicit equations.
result Only the plane and catenoid are Delaunay surfaces with nonzero constant mean curvature.
Implicit deep learning prediction rules generalize the recursive rules of feedforward neural networks. Such rules are based on the solution of a fixed-point equation involving a single vector of hidden features, which is thus only implicitly defined. The implicit framework greatly simplifies the notation of deep learni…
New framework improves robustness of implicit neural networks.
problem Ill-posedness and convergence instability in implicit neural networks.
method NEMON framework based on contraction theory for norm, including well-posedness condition, average iteration, and input-output Lipschitz constant regularization.
result Improved accuracy and robustness of implicit models with smaller input-output Lipschitz bounds.
Implicit Regularization or Implicit Conditioning? Exact Risk Trajectories of SGD in High Dimensionsstat.ML
SGD outperforms GD in high dimensions via implicit conditioning, revealed by asymptotic analysis.
problem Understanding why SGD outperforms GD in high-dimensional convex problems.
method Asymptotic analysis of multi-pass SGD on high-dimensional convex quadratics, establishing an equivalence to HSGD.
result SGD's efficiency is explained by implicit conditioning, not regularization.
IGNN captures long-range graph dependencies using fixed-point equations.
problem Limited GNN ability to capture long-range graph dependencies.
method Fixed-point equilibrium equations involving implicitly defined state vectors, leveraging Perron-Frobenius theory and projected gradient descent.
result IGNN consistently captures long-range dependencies and outperforms state-of-the-art GNNs.
New TD algorithms stabilize RL tasks by reformulating updates into fixed point equations.
problem TD learning's sensitivity to step size specification.
method Implicit TD algorithms reformulate TD updates into fixed point equations.
result Implicit TD algorithms are more stable and less sensitive to step size.
We introduce the implicit processes (IPs), a stochastic process that places implicitly defined multivariate distributions over any finite collections of random variables. IPs are therefore highly flexible implicit priors over functions, with examples including data simulators, Bayesian neural networks and non-linear tr…
The paper explores squircles and their 3D applications.
problem None explicitly stated; focuses on squircle equations and 3D surfaces.
method Examined and discussed squircle equations, then developed 3D surfaces based on these shapes.
result Developed 3D surfaces based on squircle equations.
In his book with Alan Jolis, Vers un monde sans pauvreté (1997) Yunus gives the example of a microcredit loan of 1000BDT reimbursed via 50 weekly settlements of 22BDT and correctly claims that this corresponds to the annual interest rate of 20%. But this is without taking into account that if the borrower has good reas…
Efficiently integrates stiff ODEs with vectorized methods.
problem Stiff systems and sparse training data in ODEs.
method Implicit, vectorized time integration with adjoint method.
result Achieves speed ups of greater than 100x on modern GPUs.
Noise in RNNs promotes flatter minima and more stable dynamics.
problem Understanding and optimizing the training of RNNs with noise.
method Formalizing RNNs as stochastic differential equations and analyzing the effect of noise in the hidden states.
result Noise injection in RNNs leads to flatter minima, more stable dynamics, and improved robustness.
The paper offers a framework to analyze machine learning problems using concentration of measure.
problem Analyzing machine learning algorithms defined by implicit equations.
method Develops a concentration of measure framework to solve convex problems and implicit formulations.
result Provides precise estimations for the first moments of the solution, describing the behavior and performance of machine learning classifiers.
We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit Euler scheme, an implicit Euler scheme, and a symplectic scheme.…
PANIS learns PDE surrogates for heterogeneous materials without solving the PDE.
problem Learning surrogates for parametrized PDEs in heterogeneous media.
method Physics-aware neural implicit solvers combining probabilistic learning and physics-informed discretization.
result Learned surrogates for effective solutions in heterogeneous materials without solving the reference problem.
Two effective methods for writing the dynamical equations for non-holonomic systems are illustrated. They are based on the two types of representation of the constraints: by parametric equations or by implicit equations. They can be applied to linear as well as to non-linear constraints. Only the basic notions of vecto…
HomoODE connects DEQs and Neural ODEs via homotopy continuation, improving accuracy and memory efficiency.
problem Connecting DEQs and Neural ODEs for better model performance and efficiency.
method Established a connection between DEQs and Neural ODEs using homotopy continuation, proposing HomoODE.
result HomoODE outperforms existing implicit models in accuracy and memory consumption.
We establish a glueing theorem for the Ginzburg-Landau equations in dimension . To this end, we consider a nondegenerate minimal submanifold of codimension 2, and construct a one-parameter family of solutions to the Ginzburg-Landau equations such that the energy density concentrates near this submanifold. The pr…
Implicit Q-learning and SARSA adjust step-sizes automatically, improving stability and performance.
problem Numerical instability and slow progress in Q-learning and SARSA due to step-size calibration.
method Reformulate iterative updates as fixed-point equations, scaling step-sizes inversely with feature norms.
result Implicit methods maintain stability over broader step-size ranges and achieve comparable convergence rates.
Study shows how SGD's implicit regularization relates to ridge regression.
problem Least squares regression optimization with mini-batch SGD.
method Analyzes stochastic gradient flow as a continuous-time model of SGD.
result Bound on excess risk of SGD flow over ridge regression, revealing how parameters drive risk.
The aim of this paper is mainly, after some theoretical explanations, to provide a program on Maple for computing, whatever be d, the curvature of the planar d-web implicitely defined by a differential equation F(x,y,y')=0, F being polynomial of degree d with respect to y'. Moreover, we prove in the appendix a "concent…
We study the problem to provide a triangular form based on implicit differential equations for non-linear multi-input systems with respect to the flatness property. Furthermore, we suggest a constructive method for the transformation of a given system into that special triangular shape, if possible. The well known Brun…
In this paper, we prove that the set of solutions of constraint equations for coupled Einstein and scalar fields in classical general relativity possesses Hilbert manifold structure. We follow the work of R. Bartnik [2] and use weighted Sobolev spaces and Implicit Function Theorem to prove our results.
Adam's hyperparameters implicitly regularize solutions, penalizing or impeding loss gradients' norms.
problem Implicit regularization in Adam's hyperparameters and training stage.
method Backward error analysis and ODE approximations to study Adam's behavior.
result Adam's implicit regularization depends on hyperparameters and training stage, involving different norms.
In the complex setting, let be an analytic or algebraic differential equation with -degree . We deal with the qualitative study of such equations through the geometry of the planar -web generated by the generic family of integral curves. Infinitesimal symmetries of these configurations are discu…
We consider a semilinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity which is related to a stochastic control problem with fuel constraint. The fuel constraint translates into a singular initial condition for the HJB equation. We first propose a transformation based on a change of vari…
GNIs induce asymmetric heavy-tailed noise in SGD, affecting network performance.
problem The effect of Gaussian noise injections on SGD dynamics and network performance.
method Developed a Langevin-like SDE driven by asymmetric heavy-tailed noise to model the modified SGD dynamics.
result GNIs induce an implicit bias that varies with noise heaviness and asymmetry, affecting network performance.
DEQs and explicit networks are nearly equivalent for Gaussian mixtures.
problem Understanding the equivalence between DEQs and explicit neural networks.
method Random matrix theory and analysis of kernel matrices.
result A shallow explicit network can mimic the kernel of a DEQ.
The paper challenges the belief that more inner iterations at test time improve performance in implicit deep learning.
problem The performance improvement of implicit deep learning models with increased inner iterations at test time.
method Theoretical analysis of a simple setting, validation on implicit deep learning problems.
result Overparametrization plays a key role; increasing the number of iterations at test time does not improve performance for overparametrized networks.
We describe a set of Gaussian Process based approaches that can be used to solve non-linear Ordinary Differential Equations. We suggest an explicit probabilistic solver and two implicit methods, one analogous to Picard iteration and the other to gradient matching. All methods have greater accuracy than previously sugge…
The paper analyzes the implicit bias of SGD near loss manifold and provides new insights.
problem Understanding the implicit bias of SGD near loss manifolds in overparametrized models.
method Adapting ideas from Katzenberger (1991) to analyze SGD dynamics using a stochastic differential equation (SDE).
result SGD with label noise locally decreases the sharpness of loss, leading to a global analysis of implicit bias.
A geometric flow on -forms is introduced which preserves the balanced condition of metrics, and whose stationary points satisfy the anomaly equation in Strominger systems. The existence of solutions for a short time is established, using Hamilton's version of the Nash-Moser implicit function theorem.
Study nonholonomic systems with collisions using variational principles.
problem Variational problems on nonholonomic systems with collisions.
method Extended variational principle, introduced connection on principal bundles, applied Lagrange–Poincaré–Pontryagin reduction.
result Implicit Lagrange–d'Alembert–Pontryagin equations for nonholonomic systems with collisions.
Financial Probabilities from Fisher Informationcond-mat.stat-mech
We present a novel synthesis of Fisher information and asset pricing theory that yields a practical method for reconstructing the probability density implicit in security prices. The Fisher information approach to these inverse problems transforms the search for a probability density into the solution of a differential…
Study compares methods for computing hypergradients in machine learning problems.
problem Computing exact hypergradients in machine learning is difficult.
method Investigates reverse mode iterative differentiation and approximate implicit differentiation methods.
result Unified analysis provides iteration complexity bounds and hierarchy of methods.
The purpose of this paper is to define the concept of multi-Dirac structures and to describe their role in the description of classical field theories. We begin by outlining a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are t…
For sampling from a log-concave density, we study implicit integrators resulting from -method discretization of the overdamped Langevin diffusion stochastic differential equation. Theoretical and algorithmic properties of the resulting sampling methods for and a range of step sizes are established. Ou…
Efficient numerical valuation of European options under the two-asset Kou jump-diffusion modelmath.NA
The paper efficiently solves a complex option valuation equation for two assets.
problem Valuation of European options under a two-asset Kou jump-diffusion model.
method Extends an efficient algorithm for a one-dimensional integral to a two-dimensional one, using operator splitting schemes for time discretization.
result The method achieves optimal computational cost and stable convergence for various operator splitting schemes.
We exhibit differential geometric structures that arise in numerical methods, based on the construction of Cauchy sequences, that are currently used to prove explicitly the existence of weak solutions to functional equations. We describe the geometric framework, highlight several examples and describe how two well-know…