This paper is centred on solving differential equations by symmetry groups for first order ODEs and is in response to Starrett (2007). It also explores the possibility of averting the assumptions by Olver (2000) that, in practice finding the solutions of the linearized symmetry condition is usually a much more difficul…
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48 results for “ODE Solving”
Faster training of neural ODEs using Gauß-Legendre quadrature.
problem Training neural ODEs is slow due to solving ODEs numerically.
method Use Gauß-Legendre quadrature to solve integrals faster than ODE-based methods.
result Faster training of neural ODEs, especially for large models.
Efficiently solves high-dimensional ODEs with probabilistic methods.
problem Solving high-dimensional ODEs with uncertainty quantification.
method Probabilistic numerical algorithm based on independence assumptions or Kronecker structure.
result Efficient probabilistic solutions for ODEs with millions of dimensions.
New ODE solvers improve training efficiency and accuracy.
problem Training Neural ODEs requires efficient and accurate gradient calculation.
method Presented algebraically reversible ODE solvers that are time and memory efficient, calculate exact gradients, and are numerically stable.
result Reversible solvers strictly improve upon previous architectures in efficiency and accuracy.
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.
Origin-destination (OD) matrices are often used in urban planning, where a city is partitioned into regions and an element (i, j) in an OD matrix records the cost (e.g., travel time, fuel consumption, or travel speed) from region i to region j. In this paper, we partition a day into multiple intervals, e.g., 96 15-min …
Integrating Factors and ODE Patternsphysics.comp-ph
A systematic algorithm for building integrating factors of the form mu(x,y') or mu(y,y') for non-linear second order ODEs is presented. When such an integrating factor exists, the algorithm determines it without solving any differential equations. Examples of ODEs not having point symmetries are shown to be solvable us…
We present Ordinary Differential Equation Variational Auto-Encoder (ODEVAE), a latent second order ODE model for high-dimensional sequential data. Leveraging the advances in deep generative models, ODEVAE can simultaneously learn the embedding of high dimensional trajectories and infer arbitrarily complex conti…
First-order ODEs linked to flat surfaces, leading to integrability.
problem Integrating first-order ODEs.
method Defined Riemannian metrics on variable spaces, studied surface properties, and established connections between Jacobi fields and Lie point symmetries.
result Flat associated surfaces lead to integrable first-order ODEs.
A new Fourier model improves ODE prediction.
problem Improving the accuracy of ODE solutions, especially for periodic functions.
method Constructing a Fourier state space model and a hybrid model combining Taylor and Fourier methods.
result The hybrid model can predict ODE solutions more accurately, especially for periodic functions.
The paper explores solving inverse problems for ODEs with and without constraints.
problem Understanding when second order ODEs can represent Lagrangian models with or without constraints.
method Geometric techniques to address the inverse problem for both constrained and unconstrained systems of second order ODEs.
result The constrained case presents more ambiguities and complexities than the unconstrained one.
Continuous-time MBRL framework tackles control systems with Bayesian ODEs.
problem Discretization of continuous-time systems in MBRL.
method Novel actor-critic method with Bayesian ODEs for state inference.
result Model robust against irregular and noisy data, sample-efficient, solves challenging control problems.
A simple regularization technique speeds up training of Neural ODEs.
problem Training Neural ODEs is computationally expensive.
method Randomly sampling the end time of the ODE during training.
result Significantly decreases training time and improves performance.
In the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or -distributions determined by a single function of the form , the vanishing condition for the curvature invar…
Improved neural-ODE for faster convergence and stability.
problem Stability, consistency, and convergence issues in neural-ODE solvers.
method Proposed a first-order Nesterov's accelerated gradient (NAG) based ODE-solver.
result Efficacy demonstrated in three tasks: supervised classification, density estimation, and time-series modelling.
New method combines ODE filters and numerical quadrature to propagate model uncertainty.
problem Propagation of model uncertainty in ODE solutions with uncertain parameters.
method Combining ODE filters with numerical quadrature.
result Effective propagation of both numerical and parametric uncertainty.
Solves steering problem with continuous time, Hilbert-Schmidt cost, and matrix ODEs.
problem Fixed horizon linear quadratic covariance steering in continuous time with a specific terminal cost.
method Formulates necessary conditions as a coupled matrix ODE two-point boundary value problem, designs a matricial recursive algorithm, and proves convergence.
result Proposes and proves the convergence of a matricial recursive algorithm for solving the steering problem.
The study focuses on estimating and predicting time-varying origin to destination (OD) trip tables for a dynamic traffic assignment (DTA) model. A bi-level optimisation problem is formulated and solved to estimate OD flows from pre-existent demand matrix and historical traffic flow counts. The estimated demand is then …
The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well de…
We propose a fast, simple and robust algorithm for computing shortest paths and distances on Riemannian manifolds learned from data. This amounts to solving a system of ordinary differential equations (ODEs) subject to boundary conditions. Here standard solvers perform poorly because they require well-behaved Jacobians…
Training neural ODEs on large datasets has not been tractable due to the necessity of allowing the adaptive numerical ODE solver to refine its step size to very small values. In practice this leads to dynamics equivalent to many hundreds or even thousands of layers. In this paper, we overcome this apparent difficulty b…
We accelerate CNF by reducing ODE truncation errors with polynomial regularization.
problem High computation cost of CNF due to large truncation errors in solving ODEs.
method Add polynomial regularization to approximate ODE trajectories with polynomial functions.
result 42.3% to 71.3% reduction of NFE on density estimation, 19.3% to 32.1% on variational auto-encoder.
Hypersolvers enable fast continuous-depth models for practical applications.
problem Infinite-depth models like Neural ODEs are computationally infeasible for large problems.
method Introducing hypersolvers, neural networks that solve ODEs efficiently with theoretical guarantees.
result Hypersolvers achieve comparable inference time to traditional discrete networks, making continuous-depth models practical.
This paper uses ODE to improve RNN models for time series data.
problem Improving RNN models for irregularly sampled time series data.
method Extending RNNs with Neural Ordinary Differential Equations (ODEs).
result New ODE-based RNN models reduce training and evaluation time.
DPM-Solver speeds up DPM sampling to 10-20 function evaluations.
problem Slow sampling from Diffusion Probabilistic Models (DPMs).
method Exact formulation of diffusion ODE solutions, using change-of-variable and exponentially weighted integral.
result Generates high-quality samples in 10-20 function evaluations.
Learn ODEs from noisy data using RKHS and optimization.
problem Learning nonparametric ODEs from noisy data.
method Using RKHS theory, solve a constrained optimization problem iteratively with penalty methods and Euler approximations.
result Prove a generalization bound for L2 distance between true and estimated solutions.
We solve the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a {\em complete convergent normal form} for this class of ODEs. The normal form is optimal in the sense that it is defined up to the automorphism group of the model (flat) ODE . For…
Soft-constrained PINN solves ODEs with minimal data, improving efficiency and robustness.
problem Sparse and noisy data in experiments and simulations.
method Soft-constrained Physics-informed Neural Network (PINN) with minimal labeled data.
result Soft-constrained PINN reduces need for labeled data and achieves strong generalization.
The paper solves the equivalence problem for a specific class of ODEs.
problem Solving the equivalence problem for a specific class of ordinary differential equations.
method Construction of the algebra of differential invariants for point transformations.
result The equivalence problem is solved for the given class of ODEs.
An Ode to an ODEcs.LG
New paradigm for Neural ODEs stabilizes training and improves model performance.
problem Gradient vanishing-explosion problem in training deep neural networks.
method ODEtoODE: Nested system of flows with orthogonal group constraints.
result Strong convergence results and improved downstream models in reinforcement learning and supervised learning.
Higher-order ODE solvers improve deep learning performance.
problem Improving deep learning performance using higher-order ODE solvers.
method Evaluation and improvement of Runge-Kutta (RK) methods for deep learning.
result Higher-order RK solvers can improve deep learning performance by incorporating key ingredients of optimizers.
Neural IVP solves IVPs with neural networks, overcoming scaling and conditioning issues.
problem Solving initial value PDEs with neural networks is challenging due to numerical errors and limited scalability.
method Developed an ODE-based approach to solve IVPs with neural networks, preventing ill-conditioning and scaling issues.
result Neural IVP solves challenging PDEs with neural networks efficiently and accurately.
Geometry of Third-Order Ordinary Differential Equations and Its Applications in General Relativitymath.DG
A PhD thesis written under supervision of Pawel Nurowski and defended at the Faculty of Physics of the University of Warsaw. We adress the problems of local equivalence and geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are …
New method for estimating diffusion model densities without solving flows.
problem Estimating log densities from diffusion models efficiently.
method Monte Carlo path integral estimation, avoiding flow solving.
result Significantly more scalable and efficient density estimation.
The scalar curvature equation for rotation invariant Kähler metrics on is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on in lower dimensions. We also prove that there doe…
New method combines ODE solvers with Bayesian inference for efficient model training.
problem Combining ODE solvers with Bayesian inference for efficient model training.
method Probabilistic state space model using extended Kalman filter for joint inference from differential equations and data.
result Efficient approximate Bayesian inference on latent force and ODE solution.
Theory of learning of high-dimensional controlled non-linear dynamical systems (I): models and methodscond-mat.dis-nn
Neural ODEs provide a framework for studying the training dynamics of neural networks.
problem Training dynamics of neural networks
method Dynamical mean field theory
result Derive learning curves in the high-dimensional limit
This paper compares two methods for training neural ODEs in time-series regression and CNFs.
problem Training neural ODEs for time-series regression and CNFs efficiently.
method Discretize-Optimize (Disc-Opt) vs. Optimize-Discretize (Opt-Disc) approaches.
result Disc-Opt methods can achieve similar performance as Opt-Disc at inference with drastically reduced training costs.
We introduce an effective method to solve the -harmonic forms on the Kodaira-Thurston manifold endowed with an almost complex structure and an Hermitian metric. Using the Weil-Brezin transform, we reduce the elliptic PDE system to countably many linear ODE systems. By solving a fundamental problem on line…
A rough SABR formulaq-fin.MF
Derives a rough SABR formula for short maturities.
problem Modeling volatility smiles under rough volatility.
method Derives an ODE and solves it numerically.
result Develops a very accurate approximation called the rough SABR formula.
The goal of the present paper is to propose an enhanced ordinary differential equations solver by exploitation of the powerful equivalence method of Élie Cartan. This solver returns a target equation equivalent to the equation to be solved and the transformation realizing the equivalence. The target ODE is a member of …
Parameter Estimation with Dense and Convolutional Neural Networks Applied to the FitzHugh-Nagumo ODEstat.ML
Deep neural networks solve parameter estimation for FitzHugh-Nagumo ODEs.
problem Estimating parameters of a nonlinear dynamical system from noisy time series data.
method Dense and convolutional neural networks for inverse problem solving.
result Deep neural networks accurately estimate FitzHugh-Nagumo model parameters from noisy data.
Study optimal execution in financial markets with constraints.
problem Optimal execution with non-negative constraints in a linear price impact model.
method Purely probabilistic approach via non-linear ODE.
result Complete characterization of value and optimal control.
We aim to identify the generating, ordinary differential equation (ODE) from a set of trajectories of a partially observed system. Our approach does not need prescribed basis functions to learn the ODE model, but only a rich set of Neural Arithmetic Units. For maximal explainability of the learnt model, we minimise the…
PINNs struggle with increasingly complex ODEs, especially when parameters control their complexity.
problem Evaluating physics-informed neural networks on complex coupled ODEs.
method Tuned benchmarks of partial differential equations and harmonic oscillators; varying network architecture and training method.
result PINNs fail to solve complex ODEs, revealing issues like insufficient capacity, poor conditioning, and high local curvature.
Rex solves the inverse problem for ODE/SDE solvers, improving precision and stability.
problem Inversion of ODE/SDE solvers is inaccurate and impractical for precision applications.
method Rex uses Lawson methods to convert explicit Runge-Kutta schemes into algebraically reversible ones.
result Rex achieves near-machine-precision reconstruction and improves generative models.
This paper develops an asymptotic expansion technique in momentum space for stochastic filtering. It is shown that Fourier transformation combined with a polynomial-function approximation of the nonlinear terms gives a closed recursive system of ordinary differential equations (ODEs) for the relevant conditional distri…
Differential equations (DEs) are used as numerical models to describe physical phenomena throughout the field of engineering and science, including heat and fluid flow, structural bending, and systems dynamics. While there are many other techniques for finding approximate solutions to these equations, this paper looks …