model dynamical systems

This work addresses the optimal control problem for high-dimensional, nonlinear, and analytically intractable dynamical systems—including discrete/continuous-time and deterministic/stochastic settings. We propose an end-to-end learning framework based on neural ordinary differential equations (Neural ODEs) and differentiable parameterization. The method jointly models system dynamics via Neural ODEs, represents control policies using deep neural networks, and leverages automatic differentiation and gradient-based optimization to enable implicit, differentiable parameterization of control inputs and efficient backpropagation through time. Compared to conventional numerical or analytical approaches, our framework significantly reduces computational overhead, avoids biases introduced by model simplification, and supports data-driven control under black-box dynamics. We validate its high accuracy, strong generalization, and cross-domain applicability across diverse real-world applications—including biological regulation, engineering systems, physical simulation, and medical intervention—establishing a scalable deep learning paradigm for computationally intensive dynamic system control.

Existing time-series models rely on unconstrained neural networks to model dynamics, rendering them vulnerable to distribution shifts and limiting generalization and stability. This paper reformulates time-series modeling as an optimal control problem governed by ordinary differential equations (ODEs). We propose a discrete-continuous co-architectural framework: a discrete module (e.g., RNN) generates control inputs, while a continuous module (Neural ODE) governs latent state evolution; model predictive control (MPC) enables multi-step trajectory planning and task-aware long-horizon cost minimization. By embedding control-theoretic principles, the framework guarantees system convergence—specifically, exponential convergence to the infinite-horizon optimal solution. Evaluated across multiple benchmark datasets, our method consistently outperforms state-of-the-art approaches, demonstrating superior robust generalization, enhanced out-of-distribution adaptability, and improved runtime stability.

Integrating physics-based and machine learning models for dynamic system modeling remains challenging due to difficulties in unifying multi-model representations, handling algebraic loops, and managing discontinuous events within a coherent framework. Method: This paper proposes a learnable and interpretable hybrid modeling paradigm built upon a novel wildcard architecture—the first to enable unified symbolic representation of algebraic, discrete, and differential equations—supporting end-to-end differentiable joint optimization of physics-informed and data-driven components. Grounded in systems theory and symbolic modeling principles, the approach inherently avoids algebraic loops and explicitly models discontinuities. Contribution/Results: Experiments demonstrate that the framework automatically identifies and resolves diverse dual-model compositions, achieving significant improvements over state-of-the-art methods in prediction accuracy, model interpretability, and cross-scenario generalizability.

This paper addresses discrete-time interconnected systems whose subsystem dynamics and interconnection topology are partially unknown. Method: We propose a data-driven, compositional approach to construct finite-state abstractions for formal verification and distributed controller synthesis. Subsystems are modeled individually from input-output data, and—novelly—the unknown static interconnection mapping is treated as a learnable object, enabling its symbolic abstraction. Compositionality and rigorous error propagation analysis ensure that the resulting abstraction strictly satisfies an approximate simulation relation. Contribution/Results: We theoretically establish scalability and verifiability of the abstraction. Experiments demonstrate substantial mitigation of the curse of dimensionality, enabling high-precision, low-complexity controller synthesis while preserving formal guarantees.

To address the low computational efficiency and poor accuracy of derivative computation for contact dynamics—particularly collisions and friction—in robotic simulation, which hinder convergence in reinforcement learning and trajectory optimization, this paper introduces the first unified analytical differentiation method for nonsmooth contact events. The method jointly exploits the sparsity of multibody systems and the intrinsic nonsmoothness of contact dynamics. Implemented in C++ and integrated into the Simple simulator, it supports arbitrary-degree-of-freedom rigid-body systems. On a 7-DOF manipulator and a 36-DOF humanoid robot, derivative evaluation takes only 5 μs and 95 μs, respectively—over 100× faster than state-of-the-art methods. This marks the first demonstration of microsecond-scale, high-accuracy, end-to-end differentiable robotic simulation.

This work addresses the critical influence of configuration space selection on constraint satisfaction accuracy in numerical simulations of fully constrained rigid body dynamics. From a geometric perspective, the study proposes a differential-algebraic equation (DAE) formulation and geometric integration scheme based on the Lie group SE(3). It demonstrates that when kinematic constraints correspond to subgroups of SE(3), these constraints can be preserved exactly over time. The approach elucidates the intrinsic relationship between SE(3) subgroup structures and lower-pair joints, establishing that employing SE(3) as the configuration space enables strict enforcement of constraints. This result provides both a theoretical foundation and numerical guarantees for high-fidelity simulation of rigid multibody systems.

This work addresses the challenge of synthesizing parameters for nonlinear systems under uncertain initial conditions to satisfy continuous-time Signal Temporal Logic (STL) specifications. The authors propose a novel approach that integrates gradient-based optimization with set-based reachability verification, uniquely combining learnable optimization and formal verification. This integration enables efficient exploration of high-dimensional parameter spaces while providing rigorous guarantees of robust satisfaction of STL specifications. The method is evaluated on three nonlinear systems, demonstrating both effectiveness and scalability by successfully handling parameter spaces up to 18 dimensions and delivering formally verifiable correctness assurances.

This work addresses the problem of feedback motion planning for continuous-time stochastic nonlinear systems under Signal Temporal Logic (STL) specifications by proposing a novel framework that integrates predicate erosion with probabilistic reachable tubes. Predicate erosion is employed to transform stochastic STL constraints into tightened deterministic ones, while probabilistic reachable tubes quantify the deviation of stochastic trajectories from their nominal counterparts. Leveraging contraction theory, a tracking controller is designed to establish a closed-loop planning pipeline. The proposed approach significantly reduces the conservatism inherent in conventional methods, achieving high STL satisfaction probability without compromising planning performance. Simulations and real-world experiments on a quadrupedal robot demonstrate that the method outperforms baseline approaches in both STL satisfaction rate and computational efficiency.

This work addresses the instability of conventional machine learning–based reduced-order models for stiff dynamical systems under explicit integration, a challenge exacerbated by the high computational cost and low training efficiency of implicit methods. The authors propose Trajectory-Optimized Time Reparameterization (TOTR), which formulates time remapping as an arc-length coordinate optimization problem aimed at maximizing trajectory smoothness. By minimizing the acceleration of the reparameterized trajectory, TOTR substantially enhances its learnability while enabling efficient explicit integration. Evaluated on three classes of stiff systems, the method achieves training losses one to two orders of magnitude lower than existing benchmarks and significantly improves prediction accuracy in physical time.

This work proposes a novel nonlinear dynamics modeling framework that addresses the limitations of traditional approaches—such as Euler–Lagrange equations—which are highly susceptible to measurement noise and computationally inefficient when handling nonlinear mechanical systems with external variables. By integrating a noise-robust, computationally efficient model architecture with an automated modeling pipeline, the proposed method significantly enhances inverse dynamics prediction performance. Extensive validation in representative applications, including automotive and robotic systems, demonstrates clear advantages over conventional techniques in both noise resilience and computational speed, effectively overcoming the longstanding bottleneck in modeling complex systems with strong external dependencies.