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.
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.
Neural Jump ODEs improve online filtering and classification with robust performance.
problem Online filtering and classification in settings with irregular and partial observations.
method Modeling conditional expectation using Neural Jump ODEs, with theoretical convergence guarantees.
result Demonstrated superior performance over classical methods, especially in complex scenarios.
New method uses Gaussian ODE filtering to approximate likelihoods for fast ODE inverse problems.
problem Intractable forward models in likelihood-free inference, especially for ODEs.
method Gaussian ODE filtering to construct local Gaussian likelihood approximations.
result New solvers outperform standard likelihood-free approaches on benchmark systems.
BCD-prox improves robustness and accuracy in filtering and parameter estimation.
problem Simultaneous filtering and parameter estimation of complex ODE models.
method Block coordinate descent proximal algorithm (BCD-prox) for ODE systems.
result BCD-prox outperforms state-of-the-art methods in robustness, accuracy, and training time.
New approach to ODEs using Gaussian processes and Bayesian filtering.
problem Solving ordinary differential equations (ODEs) with probabilistic methods.
method Formulate ODE solutions as Gaussian process regression problems with non-linear measurement functions.
result Developed novel Gaussian solvers with favourable stability properties.
Neural Jump ODE improves continuous-time prediction and filtering of irregularly sampled time series.
problem Theoretical guarantees for continuous-time prediction and filtering of irregularly observed time series.
method Introducing Neural Jump ODE (NJ-ODE) that models conditional expectation between observations with neural ODEs and jumps.
result Theoretical guarantees for the L2-optimal prediction are provided, showing convergence of model output to optimal prediction. 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…
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.
PROFET builds DBNs from ODEs, handling uncertainty in data and models.
problem Handling uncertainty in both data and models for DBN construction.
method Automatic DBN construction from ODEs, adaptive-time particle filtering.
result PROFET automates DBN construction and inference from ODE models.
DBNs improve accuracy of biological ODE models with missing data.
problem Uncertainty in biological ODE models with missing data.
method Converted ODE models to DBNs and used Particle Filtering for parameter estimation.
result DBNs can accurately infer model variables with missing data.
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.
A recently-introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution x and its first q derivatives \emph{a priori} as a Gauss--Markov process X, which is…
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…
Paper improves neural ODEs for forecasting non-Markovian processes.
problem Forecasting irregularly observed time series with incomplete data.
method Path-dependent Neural Jump ODEs with signature transform.
result Path-dependent NJ-ODE outperforms original framework in non-Markovian data.
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
A new method for estimating uncertainties in neural ODEs without numerical integration.
problem Accurate estimation of predictive uncertainties in neural ODEs.
method Distributional Gradient Matching (DGM) algorithm that jointly trains a smoother and a dynamics model.
result Significantly more accurate predictions compared to traditional methods.
Extends nonlinear filtering to predictable jump times.
problem Filtering with jumps in both signal and observation, especially when jump times are known.
method Derive Kushner-Stratonovich and Zakai equations for predictable discontinuities.
result Extends classical nonlinear filtering results to a setting with predictable discontinuities.
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.
Paper proposes continuous residual layers for graph neural networks.
problem Low-pass filtering effect in GCN-based models.
method Integrates Ordinary Differential Equations (ODE) to produce outputs of continuous residual layers.
result Continuous residual layers achieve better results than non-residual modules in multiple layers.
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.
Paper introduces new tasks and a model class that can solve them, using mixtures of stochastic vector fields.
problem Neural ODEs cannot solve fundamental tasks even with high-capacity vector fields.
method Proposes mixtures of stochastic vector fields and dynamic vector field selection.
result Demonstrates that the proposed model class can capture natural dynamics of human behavior.
Like many numerical methods, solvers for initial value problems (IVPs) on ordinary differential equations estimate an analytically intractable quantity, using the results of tractable computations as inputs. This structure is closely connected to the notion of inference on latent variables in statistics. We describe a …
Augmented Neural ODEs improve stability and expressiveness of ODEs.
problem Limitations of Neural ODEs in representing complex functions.
method Introducing Augmented Neural ODEs that are more expressive and stable.
result Augmented Neural ODEs outperform Neural ODEs in stability, generalization, and computational efficiency.
Generative models need per-sample confidence scores to improve quality and stability.
problem Generative models produce unreliable outputs and lack confidence measures.
method Flow Matching with Confidence (FMwC) injects noise and integrates it through the network, providing per-sample confidence scores.
result The confidence score correlates with the velocity field's divergence, offering insights into generative processes.
This study shows why training Neural ODEs is hard and proposes a new method.
problem Training Neural ODEs is challenging, especially in practice.
method Proposed a new stabilization method and provided an analytical convergence analysis.
result Insights and techniques for researchers starting work on Neural ODEs.
New model handles uneven time intervals better than traditional methods.
problem Irregularly-sampled time series data.
method Generalizes RNNs to ODE-RNNs, explicitly modeling observation gaps.
result ODE-RNNs outperform traditional models on irregular data.
Minimal surfaces in third-order ODEs identified for linear second-order ODEs.
problem Characterizing minimal surfaces in third-order ODEs.
method Analyzing submanifolds of third-order ODEs as Riemannian manifolds.
result Linear second-order ODEs with y′′=±y+β(x) are the only minimal surfaces and totally geodesic. ODE2VAE learns latent dynamics for sequential data.
problem Learning latent dynamics for high-dimensional sequential data.
method Deep generative second order ODE model with Bayesian neural networks.
result State-of-the-art performance in long-term motion prediction and imputation.
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.
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.
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.
ANODEV2 extends Neural ODEs to include evolving parameters.
problem Training and accuracy of neural networks.
method Coupled ODE-based framework for evolving neural network parameters.
result ANODEV2 achieves higher accuracy than baseline models and Neural ODEs.
Paper analyzes convergence of ODE samplers in Wasserstein distances.
problem Limited theoretical understanding of convergence properties of probability flow ODEs.
method Convergence analysis for general probability flow ODEs in 2-Wasserstein distance.
result First non-asymptotic convergence analysis for probability flow ODE samplers.
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.
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.
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.
The study identifies exceptions to fiber-preserving symmetry in ODEs and systems.
problem Identifying exceptions to fiber-preserving symmetry in ODEs and systems.
method Lie's classification of Lie algebras of vector fields, absolute and relative scalar differential invariants, conditional and vector-valued relative invariants, prolongations of actions.
result Examples of scalar ODEs and systems with symmetry groups not fiber-preserving.
New symmetry dimensions for higher order ODEs are identified.
problem Determining the maximal and submaximal symmetry dimensions for higher order ODEs.
method Cartan-geometric approach to classify symmetry dimensions.
result Next largest realizable symmetry dimensions for scalar ODEs of order ≥ 4 and vector ODEs of order ≥ 3 are determined.
The aim of this paper is to construct a Riemann-Lagrange geometry on 1-jet spaces, in the sense of d-connections, d-torsions, d-curvatures, electromagnetic d-field and geometric electromagnetic Yang-Mills energy, starting from a given linear ODEs system or a given superior order ODE. The case of a non-homogenous linear…
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.
This paper explores normalization in neural ODEs, achieving high accuracy in CIFAR-10.
problem Understanding the role of normalization in neural ODEs.
method Investigated different normalization techniques and their impact on neural ODEs performance.
result Achieved 93% accuracy in CIFAR-10 classification task.
Derives PF-ODE for infinite-dimensional functions, improving function generation tasks.
problem Efficient inference in infinite-dimensional diffusion models.
method Derives PF-ODE in infinite-dimensional function spaces.
result Reduces function evaluations while maintaining sample quality.
Neural ODEs' performance varies with numerical method, requiring adaptive step size control.
problem Neural ODEs' performance depends on the numerical method used during training.
method Proposes an adaptive step size control algorithm to ensure a valid ODE without increasing computational cost.
result Valid Neural ODEs require careful numerical method selection and step size adaptation.
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.
New vector fields integrate first-order ODEs.
problem Integrating first-order ODEs.
method Relation between Riemannian manifolds and ODEs integration.
result Integration procedure for first-order ODEs.
We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find all fundamental invariants of a system of third order ODEs and, in particular, de…
New method trains neural ODEs faster with fewer layers.
problem Training neural ODEs on large datasets is computationally expensive.
method Combines optimal transport and stability regularizations.
result Significant reductions in training time with no performance loss.