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.
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.
New method sparsifies hybrid neural ODEs for better performance and stability.
problem Excessive latent states and interactions from mechanistic models lead to training inefficiency and over-fitting.
method Automatic state selection and structure optimization combining domain-informed graph modifications with data-driven regularization.
result Improved predictive performance and robustness with desired sparsity.
We show that Neural Ordinary Differential Equations (ODEs) learn representations that preserve the topology of the input space and prove that this implies the existence of functions Neural ODEs cannot represent. To address these limitations, we introduce Augmented Neural ODEs which, in addition to being more expressive…
A new interpolation method speeds up neural ODE training.
problem Efficiently approximating gradients in neural ODEs.
method Interpolation-based technique to approximate gradients.
result Our method trains neural ODEs faster than the reverse dynamic method.
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.
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.
Generalization bounds derived for neural ODEs and deep residual networks.
problem Understanding the generalization capability of neural ODEs and deep residual networks.
method Lipschitz-based argument and analogy with deep residual networks.
result A generalization bound involving the magnitude of weight matrix differences.
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.
Neural ODEs and i-ResNet are recently proposed methods for enforcing invertibility of residual neural models. Having a generic technique for constructing invertible models can open new avenues for advances in learning systems, but so far the question of whether Neural ODEs and i-ResNets can model any continuous inverti…
Symmetry-regularized Neural ODEs improve model stability and interpretability.
problem Improving the stability and physical interpretability of Neural ODEs.
method Integrating Lie symmetries and conservation laws into the loss function.
result Symmetry-regularized Neural ODEs enhance model stability and interpretability.
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…
Neural ordinary differential equations (ODEs) have been attracting increasing attention in various research domains recently. There have been some works studying optimization issues and approximation capabilities of neural ODEs, but their robustness is still yet unclear. In this work, we fill this important gap by expl…
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.
Extends Neural ODEs to model discrete changes in continuous systems.
problem Lack of explicit termination time in existing Neural ODE formulations.
method Introduces neural event functions to implicitly define termination criteria.
result Models discrete changes in continuous systems without prior knowledge.
Balanced Neural ODEs combine VAEs and Neural ODEs for efficient time series modeling.
problem Efficiently modeling systems with time-varying inputs and varying complexity.
method Combines VAEs for dimensionality reduction and Neural ODEs for dynamics, using variational parameters to adaptively learn.
result Balanced Neural ODEs (B-NODE) efficiently approximate Koopman operator without predefined dimensionality.
MoNODEs improve neural ODEs by separating dynamic states from static factors.
problem Learning non-linear dynamics with variations across trajectories.
method Introduces time-invariant modulator variables to separate dynamic states from static factors.
result Consistently improves model generalization and far-horizon forecasting.
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.
Deep ResNets exhibit distinct scaling properties with depth, challenging neural ODE models.
problem Understanding the scaling properties of deep ResNets and their relation to neural ODEs.
method Detailed numerical experiments on weights trained by stochastic gradient descent.
result Deep ResNets can exhibit different scaling regimes, including stochastic differential equations or neither, challenging the neural ODE model.
Deep residual networks implicitly converge to neural ODEs.
problem Link between discrete and continuous deep learning models.
method Establishing implicit regularization for residual networks towards neural ODEs.
result Deep residual networks initialized as discretizations of neural ODEs converge to such ODEs during training.
Neural ODEs extended to manifolds for flexible sampling.
problem Sampling from complex multimodal distributions on non-trivial topologies.
method Extending Neural ODEs to smooth manifolds using vector fields.
result A general methodology for building normalizing flows on manifolds.
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 paper investigates how activation functions impact the training of Neural ODEs, leading to global convergence.
problem Challenges in training Neural ODEs, particularly gradient computation accuracy and convergence analysis.
method Investigates the impact of activation functions on the training dynamics of Neural ODEs.
result Establishes global convergence of Neural ODEs under gradient descent in overparameterized regimes.
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.
In this paper we prove that, in the deep limit, the stochastic gradient descent on a ResNet type deep neural network, where each layer shares the same weight matrix, converges to the stochastic gradient descent for a Neural ODE and that the corresponding value/loss functions converge. Our result gives, in the context o…
Neural Ordinary Differential Equation (Neural ODE) has been proposed as a continuous approximation to the ResNet architecture. Some commonly used regularization mechanisms in discrete neural networks (e.g. dropout, Gaussian noise) are missing in current Neural ODE networks. In this paper, we propose a new continuous ne…
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.
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
Neural ODEs control graph dynamics with low energy feedback.
problem Controlling complex dynamical systems on graphs.
method Neural Ordinary Differential Equation Control (NODEC) framework.
result NODEC learns low-energy control signals for graph dynamical systems.
It has been observed that residual networks can be viewed as the explicit Euler discretization of an Ordinary Differential Equation (ODE). This observation motivated the introduction of so-called Neural ODEs, which allow more general discretization schemes with adaptive time stepping. Here, we propose ANODEV2, which is…
Study uses machine learning to predict predator-prey dynamics without prior knowledge.
problem Predicting predator-prey interactions without prior knowledge of the system.
method Applied Neural Ordinary Differential Equations (Neural ODEs) and Universal Differential Equations (UDEs) to the Lotka-Volterra model.
result UDEs outperform Neural ODEs in predicting predator-prey dynamics, especially in noisy data.
Neural ODEs simplified using Chen-Fliess series for Rademacher complexity analysis.
problem Analyzing the complexity of neural ODE models.
method Using Chen-Fliess series to frame neural ODEs as infinite-width nets, where weights are signature of control input and features are Lie derivatives.
result Derived compact expressions for the Rademacher complexity of ODE models.
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.
RNNs are reinterpreted as kernel methods using neural ODEs.
problem Improving generalization and stability of RNNs.
method Connecting RNNs to neural ODEs and reproducing kernel Hilbert spaces.
result RNNs can be viewed as linear functions of a specific feature set.
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.
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.
A model combines GRU-D for missing data and Neural ODEs for time series continuity.
problem Challenges of informative missingness in multivariate time series data.
method Combines GRU-D for missing data imputation and Neural ODEs for temporal continuity.
result Demonstrates improved performance on a time series classification task.
New RL framework models continuous-time dynamics using neural ODEs.
problem Modeling continuous-time dynamics in semi-Markov decision processes.
method Model-based reinforcement learning with neural ODEs.
result High-performing policies developed with minimal data.
Study shows neural ODEs generalize well on synthetic graphs but struggle with degree heterogeneity and clustering.
problem Understanding neural ODEs on complex networks, especially with varying graph sizes and structures.
method Synthetic data from five dynamical systems on graphs, using Barabási-Barzel form vector fields.
result Degree heterogeneity and dynamical system type are primary factors affecting neural ODEs' generalization.
Modeling glucose distribution changes over time using neural ODEs.
problem Analyzing how continuous glucose distribution changes over time in diabetic patients.
method Combines Gaussian mixture, MMD, and Neural ODE to model temporal evolution of glucose distribution.
result Highly interpretable model detects subtle distribution shifts and remains computationally efficient.
Stochastic regularization of neural networks (e.g. dropout) is a wide-spread technique in deep learning that allows for better generalization. Despite its success, continuous-time models, such as neural ordinary differential equation (ODE), usually rely on a completely deterministic feed-forward operation. This work pr…
SONODEs and ANODEs improve learning of second order dynamics.
problem Learning dynamics governed by second order laws.
method Extended adjoint sensitivity method and theoretical analysis of ANODEs.
result SONODEs and ANODEs can learn higher order dynamics efficiently.
Improved meta-learning for dynamics using additional structured knowledge.
problem Meta-learning for dynamics with limited raw observations.
method Extended Neural ODE Process model to use privileged information.
result Improved accuracy and calibration on simulated dynamics tasks.
Investigates the relationship between ResNets and Neural ODEs, quantifying their closeness and providing training methods.
problem Quantifying the distance between ResNet dynamics and Neural ODE solutions.
method Bounding the distance between hidden state trajectories and Neural ODE solutions, using gradient descent and Heun's method.
result Gradient descent and Heun's method can implicitly regularize ResNets towards Neural ODEs, especially for smooth residual functions.
LHM integrates expert ODEs with neural ODEs for disease progression prediction.
problem Predicting disease progression under medications using limited data.
method Integrating expert-designed ODEs with machine-learned Neural ODEs.
result LHM consistently outperforms previous methods, especially in small sample regimes.
Continuous deep learning architectures have recently re-emerged as Neural Ordinary Differential Equations (Neural ODEs). This infinite-depth approach theoretically bridges the gap between deep learning and dynamical systems, offering a novel perspective. However, deciphering the inner working of these models is still a…
New model outperforms Neural ODEs while being more efficient.
problem Stable convergence and existence guarantees for implicit-depth models.
method Developed Monotone Operator Equilibrium Network (monDEQ) based on monotone operator theory.
result MonDEQ models outperform Neural ODEs and are more computationally efficient.
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.