Graph neural controlled differential equations learn graph dynamics from vertex observations.
problem Predicting future states of dynamical systems on graphs with limited vertex data.
method Incorporates graph topology information into NCDE to predict graph dynamics.
result Informed NCDE requires fewer parameters and lower MAE compared to previous methods.
Neural controlled DEs model irregular time series by adjusting based on observations.
problem Modeling irregularly sampled multivariate time series with memory-efficient adjoint-based backpropagation.
method Neural controlled differential equations (CDEs) that adjust based on subsequent observations.
result Achieves state-of-the-art performance on various datasets.
An optimal control problem associated with the dynamics of the orientation of a bipolar molecule in the plane can be understood by means of tools in differential geometry. For first time in the literature k-symplectic formalism is used to provide the optimal control problems associated to some families of partial dif…
Introduces geometric control theory for students.
problem No specific problem addressed in the abstract.
method Expository presentation of geometric control theory.
result Suitable for advanced students with solid math background.
Deep neural nets solve complex insurance math equations.
problem Optimal control problems in insurance math.
method Deep neural network algorithm for elliptic PDEs.
result Solves high-dimensional semilinear elliptic PDEs.
The paper solves optimal control problems for stochastic delay equations.
problem Optimal control of stochastic delay differential equations.
method Rewriting the problem in an infinite-dimensional Hilbert space, using dynamic programming and viscosity solutions.
result Characterizes the value function as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.
New method uses differential equations for better counterfactual analysis.
problem Estimating counterfactual outcomes for policy analysis.
method Continuous-time approach to synthetic controls using controlled differential equations.
result Improves counterfactual estimation for irregularly aligned multivariate time series.
ANCDEs improve time-series forecasting and classification using attention in NCDEs.
problem Improving time-series forecasting and classification using neural controlled differential equations.
method Integrating attention into neural controlled differential equations (ANCDEs).
result ANCDEs consistently show the best accuracy in time-series classification and forecasting.
Wave maps from circle to manifold controllable if homotopy classes match.
problem Global controllability of wave maps from circle to Riemannian manifolds.
method Characterization of controllability via homotopy classes, uniform-time global controllability between steady states, quantitative exponential stability.
result Global controllability is equivalent to homotopy class of data.
This paper presents a novel approach to numerically solve stochastic differential games for nonlinear systems. The proposed approach relies on the nonlinear Feynman-Kac theorem that establishes a connection between parabolic deterministic partial differential equations and forward-backward stochastic differential equat…
We study stochastic differential equations (SDEs) whose drift and diffusion coefficients are path-dependent and controlled. We construct a value process on the canonical path space, considered simultaneously under a family of singular measures, rather than the usual family of processes indexed by the controls. This val…
A new method models continuous-time counterfactual outcomes using neural controlled differential equations.
problem Estimating personalized healthcare outcomes over irregularly sampled data.
method Interpreting data as samples from a continuous-time process, modeling latent trajectory using controlled differential equations, and using adversarial training for time-dependent confounding.
result TE-CDE consistently outperforms existing approaches in irregularly sampled scenarios.
MSLs use parallelizable root-finding for efficient ODE and PDE solutions.
problem Efficiently solving initial value problems for ODEs and PDEs.
method Leveraging time-parallel methods, MSLs use parallelizable root-finding algorithms.
result MSLs offer significant speedups in NFEs and inference time.
The article concerns the geometrical theory of general systems Ω of partial differential equations in the \emph{absolute sense}, i.e., without any additional structure and subject to arbitrary change of variables in the widest possible meaning. The main result describes the composition series $Ω^0\subsetΩ^1\subset\cd…
The paper studies the First Order BSPDEs (Backward Stochastic Partial Differential Equations) suggested earlier for a case of multidimensional state domain with a boundary. These equations represent analogs of Hamilton-Jacobi-Bellman equations and allow to construct the value function for stochastic optimal control pro…
Novel method estimates complex nonlinear systems with stochastic differential equations.
problem Handling complex nonlinear dynamical systems with strong learning guarantees.
method Estimates drift and diffusion coefficients of continuous, multidimensional, nonlinear controlled stochastic differential equations.
result Strong theoretical guarantees including finite-sample bounds for various metrics.
Reconstructing signature features from randomized vector fields in differential equations.
problem Reconstructing signature features from controlled differential equations with random vector fields.
method Using controlled ordinary differential equations driven by continuous bounded variation curves, the study explores the extent to which signature features can be reconstructed from the non-linear flow of these equations.
result The number of signature features that can be reconstructed from the non-linear flow of controlled ordinary differential equations with random vector fields is exponential in the hidden dimension, under certain conditions.
NCDEs improve predictions for irregular time series data.
problem Theoretical understanding of NCDEs' performance and irregular time series effects.
method Combining CDE theory and neural net complexity measures.
result Generalization bound and detailed sampling and approximation bias analysis.
Study solves HJB equations for time-inconsistent control problems.
problem Time-inconsistent deterministic linear quadratic control problems.
method Characterized solutions using Riccati equations with integral terms, proving uniqueness.
result Uniqueness of solutions to equilibrium HJB equations proved.
Proves spectral inequality and null-controllability for elliptic operators on closed manifolds.
problem Proving spectral inequalities and null-controllability for elliptic pseudo-differential operators.
method Periodization approach in time inspired by global pseudo-differential calculus.
result Established spectral inequality and null-controllability for elliptic operators on closed manifolds.
Solves optimal control for stochastic processes with absorbing states.
problem Optimal control of stochastic processes with absorbing states.
method Solves through system of partial differential equations.
result Explicit solution for Merton portfolio problem with default probability.
The paper solves a complex control problem with stochastic elements and switching conditions.
problem Non-homogeneous stochastic LQ control with regime switching and random coefficients.
method Explicit optimal control and value obtained through two systems of backward stochastic differential equations (BSDEs). Existence and uniqueness of solutions proved using BMO martingales and contraction mapping method.
result Explicit optimal state feedback control and optimal value derived for the problem.
INDEQS: A Graph-Based Neural Controlled Differential Equation Framework for Forecasting
problem Forecasting time series with neural networks
method Incorporating prior knowledge of a directed graph
result Outer informedness consistently improves forecasting accuracy
We construct normed spaces of real-valued functions with controlled growth on possibly infinite-dimensional state spaces such that semigroups of positive, bounded operators (Pt)t≥0 thereon with limt→0+Ptf(x)=f(x) are in fact strongly continuous. This result applies to prove optimal rates of converge…
Novel model for predicting event intensities from static and time series data.
problem Predicting event intensities from static and irregularly sampled time series data.
method Neural controlled differential equations and signature-based CoxSig model.
result The CoxSig model provides theoretical learning guarantees and performs well on various datasets.
Predicting outcomes and planning interactions with the physical world are long-standing goals for machine learning. A variety of such tasks involves continuous physical systems, which can be described by partial differential equations (PDEs) with many degrees of freedom. Existing methods that aim to control the dynamic…
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.
Optimizes control of infectious disease spread using stochastic methods.
problem Optimizing control of highly infectious diseases like COVID-19.
method Reformulated Hamilton-Jacobi-Bellman equation as stochastic minimum principle, leading to forward-backward stochastic differential equations.
result Numerous numerical solutions presented under various scenarios.
We establish existence, uniqueness and regularity of solution results for a class of backward stochastic partial differential equations with singular terminal condition. The equation describes the value function of non-Markovian stochastic optimal control problem in which the terminal state of the controlled process is…
By the classical Martingale Representation Theorem, replication of random vectors can be achieved via stochastic integrals or solutions of stochastic differential equations. We introduce a new approach to replication of random vectors via adapted differentiable processes generated by a controlled ordinary differential …
We propose a numerical recipe for risk evaluation defined by a backward stochastic differential equation. Using dual representation of the risk measure, we convert the risk valuation to a stochastic control problem where the control is a certain Radon-Nikodym derivative process. By exploring the maximum principle, we s…
Continuous reinforcement learning such as DDPG and A3C are widely used in robot control and autonomous driving. However, both methods have theoretical weaknesses. While DDPG cannot control noises in the control process, A3C does not satisfy the continuity conditions under the Gaussian policy. To address these concerns,…
We introduce the concept of N-differential graded algebras (N-dga), and study the moduli space of deformations of the differential of a N-dga. We prove that it is controlled by what we call the N-Maurer-Cartan equation.
Neural SDEs reduce variance in stochastic simulations.
problem Efficiency of Monte Carlo simulations in finance.
method Use neural SDEs with control variates parameterized by neural networks.
result Prove optimality conditions for variance reduction in SDEs with infinite activity.
New method reduces variance in Bayesian inverse problems.
problem High variance in Monte Carlo estimates for inverse problems.
method Conditional neural control variates based on Stein's identity.
result Substantial variance reduction across different inverse problems.
Paper tackles robust control of SDEs with ambiguity, proving value function existence and applying to investment problems.
problem Robust control of SDEs with ambiguity parameters and non-Lipschitz coefficients.
method Existence and uniqueness of value function established through BSDEs with non-linear growth conditions.
result Existence and uniqueness of value function in proper space, verified through BSDEs.
This paper optimizes trading strategies to minimize risk and maximize profit while accounting for market uncertainty.
problem Optimizing trading strategies to minimize risk and maximize profit while accounting for market uncertainty.
method Relative entropy-regularized robust optimal control problem, modeled as a stochastic differential game.
result Analytical expressions for optimal strategy and trajectory are derived under specific assumptions.
New principle for optimal control with higher order differential constraints.
problem Optimal control problems with higher order differential constraints.
method Derivation of the Principle of Minimal Labour and generalization of Pontryagin Maximum Principle.
result Generalized Pontryagin Maximum Principle for higher order constraints.
Study optimal paths in Zermelo's navigation problem using geometric equations.
problem Optimal control paths in Zermelo's navigation problem.
method Geometric and differential equations approach to obtain precise ODE system.
result Obtained precise equations for optimal trajectories.
Deep learning model solves high-dimensional PDEs using Actor-Critic approach.
problem Solving high-dimensional nonlinear PDEs efficiently.
method Reformulated PDE into BSDE system, inspired by Actor-Critic algorithm for deep RL.
result Improved model with fewer parameters, faster convergence, and less hyperparameter tuning.
We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, while the Hessian is approximated by automatic differentiation of the gradient…
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.
We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we consider PDEs where the function is constrained to be positive and integrate to uni…
Neural networks solve high-dimensional HJB PDEs with asymptotic guarantees.
problem Solving high-dimensional Hamilton-Jacobi-Bellman PDEs in stochastic control theory.
method Actor-critic machine learning algorithm with a structured critic and biased gradient actor.
result The training dynamics converge to an ODE, ensuring solutions to the original problem.
Recent work has shown that reinforcement learning (RL) is a promising approach to control dynamical systems described by partial differential equations (PDE). This paper shows how to use RL to tackle more general PDE control problems that have continuous high-dimensional action spaces with spatial relationship among ac…
The goal and the main result of the paper is to provide a complete description of the field of rational differential invariants of one class of second order ordinary differential equations with scalar control parameter with respect to Lie pseudo-group of local feedback transformations. In particular, considered class d…
Paper solves time-inconsistent control problems with BSDEs.
problem Time-inconsistent stochastic control in continuous time.
method Probabilistic representation via BSDEs.
result Equilibrium value function resolved for inconsistent cases.
Paper solves complex game theory problems with new equations.
problem Zero-sum stochastic games with non-Markovian switching.
method New multidimensional SRE and BSDE solutions.
result Existence and uniqueness of SRE solutions.