Study compares 5 ODE solvers on 3 case studies, finding varying accuracy.
problem Comparing estimation accuracy of 5 ODE solvers on 3 case studies.
method Used 5 different numerical ODE solvers (Euler's, Heun's, Midpoint, Runge-Kutta 4th order, ODE45) on 3 case studies and compared their results.
result Different solvers have varying accuracy depending on the case study.
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.
New method stabilizes probabilistic ODE solvers for high accuracy.
problem Numerical instability in high-order ODE solvers.
method Accurate initialisation, coordinate change preconditioner, square-root implementation.
result Probabilistic ODE solvers can now achieve high order (up to 11) with stability.
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.
The paper improves ODE solvers by integrating diverse information types.
problem Improving accuracy and physical meaningfulness of ODE solutions.
method Leveraging probabilistic solvers to include second-order information and physical conservation laws.
result Solutions become more accurate and physically meaningful with additional information.
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.
There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al. introduced a sampling-based class of methods that are 'well-calibrated' in a specific sense. But the computational c…
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…
Developing stable and scalable probabilistic ODE solvers for stiff and high-dimensional problems.
problem Stiff and high-dimensional ODEs
method Matrix-free update step and iterative re-linearization
result Improved stability and scalability
Parallel-in-time solver reduces ODE simulation time from linear to logarithmic.
problem Efficiently solving ordinary differential equations (ODEs) with reduced computational cost.
method Formulated a parallel-in-time probabilistic numerical ODE solver using time-parallel formulation of iterated extended Kalman smoothers.
result Reduces span cost from linear to logarithmic in the number of time steps.
Likelihood-free (a.k.a. simulation-based) inference problems are inverse problems with expensive, or intractable, forward models. ODE inverse problems are commonly treated as likelihood-free, as their forward map has to be numerically approximated by an ODE solver. This, however, is not a fundamental constraint but jus…
Calibrated probabilistic solvers improve accuracy of ODE estimates.
problem Uncertainty in probabilistic ODE solutions is not well-calibrated for adaptive step sizes.
method Introduce and assess several calibration methods for probabilistic ODE solvers.
result Calibration methods interact efficiently with adaptive step-size selection, improving posteriors.
New solver avoids memory issues for long differential equations.
problem Memory constraints in adaptive probabilistic ODE solvers.
method Fixed memory demands adaptive probabilistic solver using robust state estimation.
result Eliminates memory issues for long time series simulations.
Probabilistic method combines space and time uncertainties in PDEs.
problem Separate treatment of space and time in PDE solvers obscures interactions and error quantification.
method Gaussian process interpretation of finite difference methods interacting with probabilistic ODE solvers.
result Joint quantification of space- and time-uncertainty possible without sacrificing ODE solver performance.
SONet stabilizes ODE networks for robustness without adversarial training.
problem Improving adversarial robustness of neural networks without sacrificing natural accuracy.
method SONet uses skew-symmetric ODE blocks and DOPRI5 solver for robustness.
result SONet achieves comparable robustness to adversarial defense methods without trade-off.
Paper accelerates diffusion models, improving sampling speed.
problem Low sampling speed in score-based diffusion models.
method Design of novel training-free algorithms for deterministic and stochastic samplers.
result Accelerated samplers converge faster with improved rates.
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.
Paper accelerates diffusion models without retraining, reducing evaluations.
problem Approximating target data distributions efficiently.
method Training-free sampling algorithm using high-order Lagrange interpolation.
result Requires fewer score function evaluations than previous methods.
SeqRF straightens generative model flows to speed up sampling.
problem High global truncation error in ODE-based solvers for generative models.
method SeqRF, a learning technique that straightens the probability flow.
result Significantly improved sampling speed and synthesis quality.
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…
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.
Moser Flow generates models for complex geometries on manifolds without ODE solvers.
problem Learning generative models for complex geometries like spheres and tori.
method Moser Flow is a new class of continuous normalizing flows that parameterizes the model density as the divergence of a neural network.
result Moser Flow achieves significant improvements in density estimation, sample quality, and training complexity over existing methods.
TADA improves diffusion sampling without training, up to 186% faster.
problem Efficient sampling in diffusion models, especially for high-fidelity images.
method Training-free ODE solver with higher-dimensional initial noise.
result Up to 186% faster sampling compared to state-of-the-art methods.
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.
Bayesian approach improves ODE solution accuracy.
problem Improving numerical solutions of ordinary differential equations.
method Bayesian inference with Gaussian filtering and smoothing.
result Maximum a posteriori estimate converges to true solution at polynomial rate.
DEMOTE uses neural diffusion-reaction processes to capture temporal dynamics in sparse tensor data.
problem Sparse and temporally associated tensor data with limited structural knowledge.
method Develops a neural diffusion-reaction process to estimate dynamic embeddings for tensor modes.
result Captures both commonalities and personalities in evolving tensor entries.
Gradients of neural networks can be computed efficiently for any architecture, but some applications require differential operators with higher time complexity. We describe a family of restricted neural network architectures that allow efficient computation of a family of differential operators involving dimension-wise…
In this paper we propose the use of continuous residual modules for graph kernels in Graph Neural Networks. We show how both discrete and continuous residual layers allow for more robust training, being that continuous residual layers are those which are applied by integrating through an Ordinary Differential Equation …
We study Hamiltonian Monte Carlo (HMC) for sampling from a strongly logconcave density proportional to e−f where f:Rd→R is μ-strongly convex and L-smooth (the condition number is κ=L/μ). We show that the relaxation time (inverse of the spectral gap) of ideal HMC is O(κ), improving…
We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constan…
Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability d…
Enhanced Neural ODEs outperform traditional models in image classification and video prediction.
problem Efficiently modeling time-varying dynamics in neural networks.
method Proposed a novel family of non-autonomous Neural ODEs with time-varying weights.
result Outperformed previous Neural ODE variants in speed and representational capacity.
The recently proposed Minimal Complexity Machine (MCM) finds a hyperplane classifier by minimizing an exact bound on the Vapnik-Chervonenkis (VC) dimension. The VC dimension measures the capacity of a learning machine, and a smaller VC dimension leads to improved generalization. On many benchmark datasets, the MCM gene…
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…
The paper proves neural networks are almost always surjective, impacting model safety.
problem Ensuring neural networks can generate any output, including harmful content.
method Analyzing fundamental neural architectures and generative models.
result Many neural architectures are almost always surjective, allowing for arbitrary outputs.
Paper introduces LODE-GPs for modeling data following linear ODEs.
problem Modeling data from systems of linear ODEs.
method Symbolic construction of LODE-GPs using Smith normal form algorithms.
result Improves GP modeling of data from systems of linear ODEs.
New method stabilizes GAN training by solving ODEs.
problem Stability issues in GAN training.
method Solving ordinary differential equations (ODEs) to stabilize GAN training.
result Well-known ODE solvers can stabilize GAN training.
Survival MDN uses invertible functions to speed up survival analysis models.
problem Training neural ODEs for survival analysis is computationally expensive.
method Survival MDN applies an invertible positive function to MDN outputs.
result Survival MDN outperforms or matches other models on concordance, Brier score, and log-likelihood.
Continuous Normalizing Flows (CNFs) have emerged as promising deep generative models for a wide range of tasks thanks to their invertibility and exact likelihood estimation. However, conditioning CNFs on signals of interest for conditional image generation and downstream predictive tasks is inefficient due to the high-…
DALTON improves ODE parameter estimation by learning from noisy data.
problem High sensitivity to parameters in ODEs produces unreliable parameter estimates.
method Data-adaptive probabilistic likelihood approximation for ODEs.
result DALTON produces more accurate parameter estimates than existing methods.
We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions. This is achieved by defining the measurement sequence to consist of the observations of the difference between the derivative …
Researchers develop neural optimal transport with Lagrangian costs for efficient computation.
problem Optimal transport between measures with Lagrangian costs for systems with geometric constraints.
method Neural network approach to compute geodesics and optimal transport maps efficiently.
result Efficient computation of geodesics and optimal transport maps without ODE solvers.
Paper introduces a neural framework for accurate energy forecasting.
problem Challenges of forecasting energy demand and supply due to variability of renewable sources and dynamic consumption patterns.
method Integrates Neural ODEs, graph attention, multi-resolution wavelet transformations, and adaptive learning of frequencies.
result Consistently outperforms state-of-the-art baselines in various forecasting metrics across diverse datasets.
New bounds for generative models under weaker assumptions.
problem Establishing convergence guarantees for generative models under weak assumptions.
method Non-asymptotic 2-Wasserstein distance bounds for probability flow ODEs under weak log-concavity and Lipschitz continuity.
result Concrete convergence rates for generative models, including non-log-concave distributions.
LFlows model fluid densities and velocities using invertible maps that satisfy the continuity equation.
problem Modeling fluid densities and velocities continuously in space and time.
method LFlows are based on invertible maps that satisfy the continuity equation, derived from classical theory of Lagrangian flows for smooth vector fields.
result LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D.
We present a derivation and theoretical investigation of the Adams-Bashforth and Adams-Moulton family of linear multistep methods for solving ordinary differential equations, starting from a Gaussian process (GP) framework. In the limit, this formulation coincides with the classical deterministic methods, which have be…
New RNN model handles long-term dependencies in irregularly-sampled time series.
problem Handling long-term dependencies in irregularly-sampled time series data.
method Designing ODE-LSTMs that separate memory from continuous-time state.
result ODE-LSTMs outperform other RNN-based models on non-uniformly sampled data with long-term dependencies.
Flow Matching enables robust training of CNFs with various probability paths.
problem Training Continuous Normalizing Flows (CNFs) at large scales.
method Flow Matching (FM) is a simulation-free approach for training CNFs by regressing vector fields of conditional probability paths.
result Flow Matching with diffusion paths yields more robust and stable training compared to diffusion-based methods.