Lagrangian Neural Networks

In this paper, we propose the Dissipative Lagrangian Neural Network (dLNN), a continuous-time system identification technique that separately learns the conservative and dissipative dynamics of a mechanical system directly from data. A key contribution is a novel parametrisation of dissipative forces that (i) is structurally constrained to be dissipative, enforcing passivity of the learned model, (ii) cannot absorb conservative mechanics, and (iii) retains sufficient expressivity to represent a broad family of state-dependent damping and regularised friction forces. The model is validated on prediction tasks on two real-world mechanical systems: trajectory prediction of a chaotic double pendulum and inverse-dynamics modelling of a six-degree-of-freedom industrial robot. Across these tasks, the dLNN demonstrates improved predictive performance over existing Lagrangian architectures while remaining competitive with a black-box neural model. The imposed physical structure also improves generalisation to unseen states, with the benefits becoming particularly pronounced when training data is limited. Finally, we introduce a symbolic distillation procedure that extracts closed-form expressions from the learned physical components. Applied to the industrial robot, this produces interpretable symbolic models that retain high predictive accuracy while reducing the computational overhead associated with evaluating neural Lagrangian models.

Lagrangian Neural Networks (LNNs) learn mechanical systems from trajectory data by parameterizing a scalar Lagrangian and deriving the dynamics through the Euler-Lagrange equations, biasing the model toward motion that respects the variational structure of mechanics. The interactions studied so far have been pairwise-additive, leaving open whether LNNs can recover an irreducible many-body interaction from motion alone. We address this using the Axilrod-Teller triple-dipole interaction, the leading three-body dispersion term, whose dependence on the collective geometry of three particles cannot be represented by any sum of pairwise terms. We prescribe the kinetic energy exactly and learn only the potential, as a function of interparticle distances and angles. Combined with a signal-aware sampling strategy that concentrates training data where the three-body signal is strongest while preserving broad coverage of configuration space, this recovers the interaction from accelerations alone. The learned potential reproduces the radial scaling, angular dependence, and characteristic sign reversal of the Axilrod-Teller interaction, and generates stable trajectories. Embedding it within a pairwise Lennard-Jones potential lets us examine its observability: the three-body term is recovered once its dynamical signature exceeds the residual modeling error, which it does cleanly at amplified coupling but not at the coupling of a real noble gas, bracketing the transition between the two. Finally, symbolic regression on the trained potential, with no assumed functional form, returns the Axilrod-Teller expression with angular numerator, radial law, and coupling constant intact. An LNN with domain-specific structure can thus recover an irreducible three-body interaction from trajectories as a closed-form law, pointing toward learning interaction laws whose structure is not known a priori.

Lagrangian Neural Networks (LNNs) learn dynamics from trajectory data by parameterizing a scalar Lagrangian and deriving accelerations through the Euler--Lagrange equation, providing a variational inductive bias that improves physical consistency over purely data-driven models. LNNs have recently been extended to general relativistic settings, leaving Schwarzschild geodesics as an open challenge. Training on complex relativistic systems is hindered by the diversity of the dynamics, as well as numerical instabilities and the difficulty of adequately sampling physically relevant, diverse orbital regimes. We introduce two coordinated improvements: a decoupled position--velocity normalization scheme that independently rescales each coordinate and its time derivatives, reweighting acceleration targets to balance contributions across all channels; and an energy-constrained log-uniformly sampled dataset generation strategy informed by the innermost stable circular orbit (ISCO) structure. With these improvements, our model accurately reproduces perihelion precession and relativistic time dilation, conserves energy and angular momentum to within a few percent even after completing multiple orbits around the central mass and predicts orbital trajectories across eight orbital scenarios in close, medium, and far-field regimes. These results establish that LNNs, augmented with physically motivated normalization and sampling, can reliably learn geodesic dynamics in curved spacetime. Our work also represents the first successful training of LNNs on Schwarzschild geodesics and offers a promising direction for data-driven modeling of complex relativistic systems.

This work proposes a latent Lagrangian-based framework for reduced-order modelling of forced nonlinear dynamical systems. In contrast with conventional Lagrangian or Hamiltonian neural networks, our approach learns a set of latent coordinates sufficient to capture the dynamics conjointly with two neural networks for the latent kinetic and latent potential energies, and leverages force supervision to eliminate the need for an ODE solver during training. Consistency of physical laws in the latent space is ensured through the principle of virtual work. Results show that the model effectively learns the subtle dynamics induced by the system's nonlinearity and non-convex potential energy, and generalizes well to unseen forces and initial conditions. These observations confirm the physical relevance of the proposed approach, and its interest for model reduction.

We investigate the applicability of Lagrangian and Hamiltonian Neural Network models to a dissipative system that has explicit time dependence in its Lagrangian, Hamiltonian, and total energy. To do so we consider these neural network models for simulated systems of a harmonic one-dimensional, one-component oscillator with damping, as well as without damping for comparison. We find that both the Lagrangian and Hamiltonian approaches are able to predict the empirical physical behavior of the damped oscillator systems and to effectively ``learn'' to varying degrees the underlying Lagrangians and Hamiltonians, as has previously been shown to be the case with undamped oscillator systems. These investigations elucidate important properties of Lagrangian and Hamiltonian mechanics, including properties that are not manifest when considering systems without explicit time dependence.

Task-agnostic assistive exoskeleton control based on human intention offers greater flexibility than conventional approaches that rely on predefined tasks or motion patterns. Human joint torque estimation enables task-agnostic assistance by characterizing user actions. Physics-consistent methods such as Deep Lagrangian Networks (DeLaN) have been applied to estimate the human torques in multi-user settings, but existing approaches cannot adapt to a specific user without retraining, and do not account for intermittent contacts during locomotion. We propose ExoLaN, a Context-Aware DeLaN for human-exoskeleton interaction that learns the full coupled system dynamics while adapting to changes in interaction context. ExoLaN combines temporal context with partial contact-force measurements from force-sensitive insoles to infer latent dynamics embeddings and estimate generalized contact torques. On seven unseen users performing 21 unseen tasks, ExoLaN reduces torque estimation MSE by 7% compared to a black-box baseline. Beyond inverse dynamics, ExoLaN serves as a unified model that also enables accurate forward prediction: training with a multi-step prediction loss reduces acceleration MSE by 59% and long-horizon position and velocity errors by 60% and 93%, respectively, compared with a single-step loss. Moreover, the learned latent context captures task information without explicit task labels, making it a promising signal for task-aware assistive control.

Forecasting the long-horizon evolution of mechanical systems from position-only observations is a pivotal yet difficult task, as hidden velocities and trajectory-specific physical properties must be inferred simultaneously. Although physics-guided neural networks like Lagrangian Neural Networks (LNNs) guarantee physical plausibility, they generally require complete state inputs and lack adaptability to changing system parameters. To break these limitations, we introduce History-informed Lagrangian Neural Networks (HiLNN). Grounded in the insight that temporal position sequences implicitly encode underlying dynamics, HiLNN employs a recurrent encoder to extract a latent context from history. This context not only reconstructs the unobserved initial velocity but also adaptively modulates the mass matrix, potential energy, and damping coefficients of a structured Lagrangian system. By leveraging a differentiable RK4 rollout scheme, the entire pipeline is optimized end-to-end under multi-step trajectory supervision and energy-consistency regularization. Empirical evaluations across conservative, dissipative, and heterogeneous variable-parameter systems show that HiLNN delivers superior long-term prediction accuracy and maintains precise energy profiles compared to state-of-the-art baselines. The source code is publicly available at https://github.com/yingtian22/History-informed-LNN.

This paper proposes a Bayesian filtering-based approach for learning the dynamics of a physical system from partial, noisy measurements. We model the system dynamics using a Lagrangian mechanics formulation. As in Lagrangian neural networks (LNNs), we parameterize the kinetic and potential energies with neural networks. The unknown external forces in the Lagrangian formulation are modeled as white Gaussian noise. The corresponding Euler--Lagrange equations then yield a continuous-time stochastic state-space model (SSM) that describes the system dynamics. The neural network parameters and system states are then jointly learned via a maximum-likelihood method using Gaussian-approximation-based Bayesian filters. The effectiveness of the proposed method is demonstrated on pendulum and Duffing oscillator examples, and its performance is compared with conventional LNNs and with approximate Bayesian filters using known system models.

We present the first method to directly use a learned continuous Lagrangian to forecast the dynamics of systems governed by partial differential equations, exploiting the inherent conservative structure to achieve stable long-range predictions. We develop an optimization-based integrator that minimizes the squared Euler--Lagrange residual via a mesh-free near-symplectic construction on local space-time patches. Different from integrators for analytical models, integrators for learned models should decouple model error (phase error) from integration error (conservation error). By relying on optimization rather than time-stepping, we bypass the global coupling inherent to fixed discretizations, which slows time- and space-stepping and complicates learning. Our method scales linearly with domain size via Jacobi iteration, and places no structural requirements on the learned network, allowing it to be coupled with existing physics-guided machine learning (ML) methods. We validate our approach on a learned representation of a double pendulum, a one-dimensional wave equation, and a two-dimensional wave equation. Our method achieves error comparable to classical symplectic methods while generalizing to spatially varying dynamics and arbitrary boundary conditions without retraining.

State of the art quadrupedal locomotion approaches integrate Model Predictive Control (MPC) with Reinforcement Learning (RL), enabling complex motion capabilities with planning and terrain adaptive behaviors. However, they often face compounding errors over long horizons and have limited interpretability due to the absence of physical inductive biases. We address these issues by integrating Lagrangian Neural Networks (LNNs) into an RL MPC framework, enabling physically consistent dynamics learning. At deployment, our inverse dynamics infinite horizon MPC scheme avoids costly matrix inversions, improving computational efficiency by up to 4x with minimal loss of task performance. We validate our framework through multiple ablations of the proposed LNN and its variants. We show improved sample efficiency, reduced long-horizon error, and faster real time planning compared to unstructured neural dynamics. Lastly, we also test our framework on the Unitree Go1 robot to show real world viability.

Robotic systems operating in unstructured environments must operate under significant uncertainty arising from intermittent contacts, frictional variability, and unmodeled compliance. While recent model-free approaches have demonstrated impressive performance, many deployment settings still require predictive models that support planning, constraint handling, and online adaptation. Analytical rigid-body models provide strong physical structure but often fail to capture complex interaction effects, whereas purely data-driven models may violate physical consistency, exhibit data bias, and accumulate long-horizon drift. In this work, we propose STRIDE, a dynamics learning framework that explicitly separates conservative rigid-body mechanics from uncertain, effectively stochastic non-conservative interaction effects. The structured component is modeled using a Lagrangian Neural Network (LNN) to preserve energy-consistent inertial dynamics, while residual interaction forces are represented using Conditional Flow Matching (CFM) to capture multi-modal interaction phenomena. The two components are trained jointly end-to-end, enabling the model to retain physical structure while representing complex stochastic behavior. We evaluate STRIDE on systems of increasing complexity, including a pendulum, the Unitree Go1 quadruped, and the Unitree G1 humanoid. Results show 20% reduction in long-horizon prediction error and 30% reduction in contact force prediction error compared to deterministic residual baselines, supporting more reliable model-based control in uncertain robotic environments.

Model-based reinforcement learning (MBRL) is sample-efficient but depends on the accuracy of the learned dynamics, which are often modeled using black-box methods that do not adhere to physical laws. Those methods tend to produce inaccurate predictions when presented with data that differ from the original training set. In this work, we employ Lagrangian neural networks (LNNs), which enforce an underlying Lagrangian structure to train the model within a Dyna-based MBRL framework. Furthermore, we train the LNN using stochastic gradient-based and state-estimation-based optimizers to learn the network's weights. The state-estimation-based method converges faster than the stochastic gradient-based method during neural network training. Simulation results are provided to illustrate the effectiveness of the proposed LNN-based Dyna framework for MBRL.

Physics-informed deep learning models have emerged as powerful tools for learning dynamical systems. These models directly encode physical principles into network architectures. However, systematic benchmarking of these approaches across diverse physical phenomena remains limited, particularly in conservative and dissipative systems. In addition, benchmarking that has been done thus far does not integrate out full trajectories to check stability. In this work, we benchmark three prominent physics-informed architectures such as Hamiltonian Neural Networks (HNN), Lagrangian Neural Networks (LNN), and Symplectic Recurrent Neural Networks (SRNN) using the DeepChem framework, an open-source scientific machine learning library. We evaluate these models on six dynamical systems spanning classical conservative mechanics (mass-spring system, simple pendulum, double pendulum, and three-body problem, spring-pendulum) and non-conservative systems with contact (bouncing ball). We evaluate models by computing error on predicted trajectories and evaluate error both quantitatively and qualitatively. We find that all benchmarked models struggle to maintain stability for chaotic or nonconservative systems. Our results suggest that more research is needed for physics-informed deep learning models to learn robust models of classical mechanical systems.