Signature tensors uniquely identify ODE solutions.
problem Identifying ODE solutions from signature tensors.
method Geometric theory of nonlinear systems of ODEs.
result Necessary and sufficient algebraic conditions for signature tensors to represent ODE solutions.
WENDy now estimates nonlinear ODEs with noisy data.
problem Estimating parameters of nonlinear ODEs with noisy data.
method WENDy-MLE algorithm for maximum likelihood estimation of nonlinear-in-parameters ODEs.
result WENDy-MLE outperforms other methods in accuracy, speed, and domain of convergence.
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 (2,3,5)-distributions determined by a single function of the form F(q), the vanishing condition for the curvature invar…
Dual optimization connects ERM-fDR to normalization function.
problem Empirical risk minimization with f-divergence regularization.
method Dual formulation, Legendre-Fenchel transform, implicit function theorem, nonlinear ODE.
result Computational method to calculate normalization function efficiently.
A novel model uses ODE-based random features to model nonlinear dynamical systems.
problem Modeling highly nonlinear dynamical systems with uncertainty quantification.
method Compositions of physics-informed random features derived from ODEs, combined with deep Gaussian processes and approximate Bayesian inference.
result The model effectively captures nonlinear behavior in real-world multivariate time series data and achieves comparable performance to other models on benchmark tasks.
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.
Study finds multiple periodic solutions to ODEs related to curvature problems.
problem Finding multiple positive periodic solutions to quasilinear ODEs.
method Global bifurcation techniques applied to second order quasilinear ODEs.
result Bifurcation-theoretic proof of nonuniqueness for conformal metrics with constant scalar curvature.
Paper addresses identifiability and asymptotics of ODE systems from noisy data.
problem Identifying parameters and causal structure of linear ODE systems from discrete observations.
method Developed sufficient conditions for identifiability, proved consistency and asymptotic normality of NLS estimator, constructed confidence sets, and inferred causal structure.
result Consistent and asymptotically normal parameter estimator for linear ODE systems under mild conditions.
This paper tackles Bayesian system identification with probabilistic numerical methods.
problem Accurately modeling nonlinear dynamic systems from noisy data.
method Probabilistic Sequential Monte Carlo (SMC) combined with probabilistic numerical integration.
result Efficient identification of latent states and system parameters from noisy measurements.
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…
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.
GrADE uses graph neural networks and Neural ODE for solving time-dependent nonlinear PDEs efficiently.
problem Solving time-dependent nonlinear PDEs is computationally challenging and time-consuming.
method GrADE combines graph neural networks for spatial modeling and Neural ODE for temporal modeling, using attention mechanisms.
result GrADE efficiently solves PDEs, demonstrating scalability and better accuracy compared to existing methods.
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.
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.
In this work, we investigate the problem of finding surfaces in the Lorentz-Minkowski 3-space with prescribed skew (S) and mean (H) curvatures, which are defined through the discriminant of the characteristic polynomial of the shape operator and its trace, respectively. After showing that H and S can be interpr…
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.
Study Nash competition among dealers quoting prices to clients with unknown trading motives.
problem Adverse selection and inventory costs in dealer-client interactions.
method Analyzes one-shot Nash competition with unknown client type and inventory constraints.
result Unique symmetric Nash equilibrium exists and can be characterized by a nonlinear ODE.
We introduce a flexible, scalable Bayesian inference framework for nonlinear dynamical systems characterised by distinct and hierarchical variability at the individual, group, and population levels. Our model class is a generalisation of nonlinear mixed-effects (NLME) dynamical systems, the statistical workhorse for ma…
Gradient matching with Gaussian processes is a promising tool for learning parameters of ordinary differential equations (ODE's). The essence of gradient matching is to model the prior over state variables as a Gaussian process which implies that the joint distribution given the ODE's and GP kernels is also Gaussian di…
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.
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.
Letter analyzes training dynamics of a nonlinear contrastive learning model in high dimensions.
problem Understanding training dynamics of nonlinear contrastive learning models in high-dimensional settings.
method High-dimensional analysis using McKean-Vlasov PDEs and low-dimensional ODEs.
result The model's performance evolves according to specific ODEs, revealing features like feature learnability and noise effects.
NP-ODE models FEA simulations with uncertainty, improving accuracy and efficiency.
problem Limitations of FEA in terms of computational cost and uncertainty quantification.
method Physics-informed neural process aided ordinary differential equations (NP-ODE).
result NP-ODE outperforms benchmark methods in uncertainty quantification and prediction accuracy.
New symmetries found for scalar and vector ODEs of arbitrary dimensions.
problem Identifying symmetries for scalar and vector ODEs of arbitrary dimensions.
method Explicit expressions and abelian Lie algebra for non-Cartan symmetries in arbitrary dimensions.
result Non-Cartan symmetries characterize linearizable systems of ODEs but not nonlinear ones.
The aim of this paper is to construct a natural Riemann-Lagrange differential geometry on 1-jet spaces, in the sense of nonlinear connections, generalized Cartan connections, d-torsions, d-curvatures, jet electromagnetic fields and jet electromagnetic Yang-Mills energies, starting from some given nonlinear evolution OD…
To understand the fundamental trade-offs between training stability, temporal dynamics and architectural complexity of recurrent neural networks~(RNNs), we directly analyze RNN architectures using numerical methods of ordinary differential equations~(ODEs). We define a general family of RNNs--the ODERNNs--by relating t…
This paper constructs GCM hypersurfaces in Kerr spacetimes.
problem Extending the Kerr family stability proof to full stability.
method Concatenating a 1-parameter family of GCM spheres by solving an ODE system.
result Removes symmetry restrictions in GCM procedure.
Investigates optimal portfolio selection with regime-switching-induced stock price shocks.
problem Mean-variance portfolio selection with regime-switching and stock price jumps.
method Modeling regime-switching and stock price jumps, deriving optimal portfolio strategy and efficient frontier using ODEs.
result Added complexity due to regime-switching-induced stock price shocks, leading to nonlinear ODEs.
We study a class of nonlinear pricing models which involves the feedback effect from the dynamic hedging strategies on the price of asset introduced by Sircar and Papanicolaou. We are first to study the case of a nonlinear demand function involved in the model. Using a Lie group analysis we investigate the symmetry pro…
Augmented KRnet improves flow-based generative modeling by maintaining exact invertibility.
problem Maintaining exact invertibility in flow-based generative models.
method Integrates augmented dimensions into KRnet to achieve full nonlinear updates in two iterations, keeping exact invertibility.
result Augmented KRnet achieves full nonlinear updates in two iterations, maintaining exact invertibility.
We study five dimensional geometries associated with the 5-dimensional irreducible representation of GL(2,R). These are special Weyl geometries in signature (3,2) having the structure group reduced from CO(3,2) to GL(2,R). The reduction is obtained by means of a conformal class of totally symmetric 3-tensors. Among all…
Ancient solutions arise in the study of Ricci flow singularities. Motivated by the work of Fateev on 3-dimensional ancient solutions we construct high dimensional ancient solutions to Ricci flow on spheres and complex projective spaces as well as the twistor spaces over a compact quaternion-Kahler manifold. Differing f…
A new method combines ANN and Laplace for fast Bayesian inference in ODE models.
problem Bayesian inference for ODE systems with non-analytical solutions is computationally expensive.
method Hybrid approach using ANN for tractable likelihood and Laplace approximation.
result Effective posterior inference with improved computational cost compared to traditional methods.
Bayesian Neural ODEs improve vessel trajectory prediction with better uncertainty estimates.
problem Challenges in predicting vessel trajectories from irregular AIS data.
method Adopted a Gaussian process (GP) kernel-based prior on the vector field evaluated at measurement points, combined with probabilistic multiple shooting for long trajectories.
result Improved accuracy and uncertainty quantification in vessel trajectory predictions.
Graph-Coupled Oscillator Networks (GraphCON) tackles graph-based learning problems.
problem The oversmoothing problem in Graph Neural Networks (GNNs).
method GraphCON is a novel framework based on discretizations of ODEs modeling oscillators coupled via graph adjacency.
result GraphCON mitigates the oversmoothing problem and exploding/vanishing gradients issues.
Parameter identification and comparison of dynamical systems is a challenging task in many fields. Bayesian approaches based on Gaussian process regression over time-series data have been successfully applied to infer the parameters of a dynamical system without explicitly solving it. While the benefits in computationa…
Study proves fluid limits of fragmented limit-order markets.
problem Modeling fragmented limit-order markets with small and frequent orders.
method Proved convergence of discrete system to fluid limit characterized by coupled nonlinear ODEs.
result Fluid system converges to stationary equilibrium state over time.
Statistical methods remain relevant for ODE inverse problems, especially with sparse data.
problem The relevance of statistical methods in the era of deep learning for ODE inverse problems.
method Employed physics-informed neural networks (PINN) and manifold-constrained Gaussian process inference (MAGI) to compare statistical and deep learning approaches.
result Statistically principled methods outperform deep learning models in tasks like parameter inference and trajectory reconstruction.
Investigates portfolio selection for rank-dependent utilities in incomplete markets.
problem Portfolio selection for agents with rank-dependent utility in incomplete financial markets.
method Characterizes deterministic strict equilibrium strategies for constant-coefficient and time-invariant probability weighting functions. Addresses the issue of selecting an optimal strategy from multiple equilibrium strategies for time-variant probability weighting functions.
result Characterizes deterministic strict equilibrium strategies and identifies optimal strategies from multiple equilibrium strategies.
EFiGP uses Fourier and eigen-decomposition for efficient ODE parameter estimation.
problem Parameter estimation and trajectory reconstruction for noisy, sparse, nonlinear ODE systems.
method EFiGP integrates Fourier transformation and eigen-decomposition into a physics-informed Gaussian Process framework.
result EFiGP efficiently estimates ODE parameters and recovers trajectories from noisy data.
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.
LiLaN uses linear latent networks to solve stiff ODEs efficiently.
problem Solving stiff ordinary differential equations (StODEs) requires expensive methods.
method LiLaN integrates latent dynamics analytically, avoiding explicit/implicit integration.
result LiLaN can approximate stiff nonlinear systems to any accuracy epsilon.
DLFM models complex systems with uncertainty, outperforming traditional methods.
problem Modeling highly nonlinear dynamical systems with robust uncertainty quantification.
method Deep latent force model (DLFM) using physics-informed kernels derived from ODEs.
result DLFM achieves comparable performance to non-physics-informed models on univariate tasks and captures dynamics in real-world data.
REGS samples from unnormalized distributions using gradient flow and neural networks.
problem Sampling from unnormalized distributions with high accuracy and efficiency.
method REGS is a particle method that iteratively transforms samples from a reference distribution to match an unnormalized target distribution using Wasserstein gradient flow and neural networks.
result REGS outperforms state-of-the-art methods in sampling from challenging multimodal distributions and real datasets.
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.
Novel deep learning model improves AUC prediction for tacrolimus dosing.
problem Model misspecification in current pharmacokinetic models.
method Latent Neural-ODE for learning individualized pharmacokinetic dynamics.
result Latent ODE model outperforms standard methods in AUC prediction accuracy.
A new method estimates parameters of complex models using ordinary least squares.
problem Estimating parameters of nonlinear dynamic models from time series data.
method Physics-Informed Regression (PIR) using regularized ordinary least squares.
result PIR outperforms physics-informed neural networks (PINN) in parameter estimation.
In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed fo…