PINODE: Hybrid Physics-Informed Neural ODE Model

• PINODE framework is a hybrid method that combines classical Lagrangian mechanics with neural ODEs to model physical systems.
• It partitions dynamics into analytically known components and learned non-conservative forces using a neural network.
• Empirical evaluations on a cart-pole system demonstrate improved accuracy, data efficiency, and model interpretability.

The Physics-Informed Neural Ordinary Differential Equation (PINODE) framework is an approach for modeling the dynamics of complex physical systems by merging classical Lagrangian mechanics with neural network-based function approximation. PINODE is specifically designed for scenarios where a significant portion of system dynamics can be physically derived, but hard-to-model effects—such as non-conservative forces—require expressive learning from data. This methodology enables practitioners to retain interpretability and physical plausibility while achieving modeling flexibility and accuracy in engineering, robotics, and control contexts (Roehrl et al., 2020).

1. Theoretical Foundation and Motivation

PINODE is motivated by the limitations of both purely first-principles modeling and fully data-driven approaches. Traditional analytic methods provide bias but struggle with uncertain or unmodeled dynamics, while black-box neural networks have high variance, require large datasets, and often lack interpretability.

PINODE adopts a hybrid paradigm. The mechanical system under consideration is described via generalized coordinates $q \in \mathbb{R}^n$ and controlled with input $u$. Its motion is prescribed by the Lagrangian:

$L(q, \dot{q}) = T(q, \dot{q}) - V(q)$

where $T$ and $V$ are the kinetic and potential energy, respectively. The system is further modeled using the Euler–Lagrange equation, incorporating non-conservative generalized forces $Q^{\mathrm{ncons}}(q, \dot{q}, u)$:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = Q^{\mathrm{ncons}}(q, \dot{q}, u)$$

After reduction, the system's second-order equations take the familiar form:

$$M(q)\, \ddot{q} + C(q, \dot{q})\, \dot{q} + G(q) = Q^{\mathrm{ncons}}(q, \dot{q}, u)$$

where $M(q)$ is the inertia matrix, $C(q, \dot{q})$ contains Coriolis and centrifugal effects, and $u$0 is the conservative force vector.

2. Hybrid Modeling: Separation of Physically Known and Unknown Dynamics

A core principle of PINODE is partitioning the dynamics into analytically known and unknown components:

• The terms $u$1, $u$2, and $u$3 are explicitly modeled based on available system parameters (e.g., masses, link lengths, gravitational constants).
• The non-conservative force $u$4—which may encompass Coulomb friction, stick-slip effects, aerodynamic drag, or other hard-to-model phenomena—is approximated by a neural network:

$u$5

with $u$6 the network's trainable weights.

Thus, the acceleration update is:

$u$7

This separation enables PINODE to incorporate prior physical knowledge while adapting to unmodeled complexities.

3. Neural ODE Formulation and Integration

PINODE represents the full dynamical model as a first-order ODE over the state $u$8: $u$9 where: $L(q, \dot{q}) = T(q, \dot{q}) - V(q)$0 Numerical integration (e.g., with a 4th-order Runge–Kutta scheme) lets the entire system be differentiable end-to-end, making it suitable for optimization via gradient-based methods. All physics-based and learned terms are part of the computational graph, ensuring accurate and physically plausible time evolution during learning and inference.

4. Training Procedure and Loss Function

Training occurs on recorded state/input trajectories $L(q, \dot{q}) = T(q, \dot{q}) - V(q)$1. Starting from $L(q, \dot{q}) = T(q, \dot{q}) - V(q)$2, PINODE integrates the ODE for one time step to produce the prediction $L(q, \dot{q}) = T(q, \dot{q}) - V(q)$3. The loss minimized is: $L(q, \dot{q}) = T(q, \dot{q}) - V(q)$4 where $L(q, \dot{q}) = T(q, \dot{q}) - V(q)$5 and $L(q, \dot{q}) = T(q, \dot{q}) - V(q)$6 set the trade-off between position and velocity error. Gradients are computed by backpropagation through the ODE integrator, enabling efficient optimization (e.g., with Adam). Only the network parameters $L(q, \dot{q}) = T(q, \dot{q}) - V(q)$7 associated with the residual non-conservative forces are updated, maintaining the interpretability of the analytic components.

5. Implementation and Experimental Evaluation

In empirical evaluation, PINODE was applied to a physical cart-pole testbed—a standard benchmark in dynamics and control. The setup comprised a pendulum of length $L(q, \dot{q}) = T(q, \dot{q}) - V(q)$8 m and mass $L(q, \dot{q}) = T(q, \dot{q}) - V(q)$9 kg attached to a cart of mass $T$0 kg; both cart and pendulum states, as well as control inputs, were measured at 50 Hz.

A four-layer MLP was used for $T$1, taking $T$2 as input, with two hidden layers of 50 ReLU units each and a final tanh scalar output (totaling 5,451 parameters). Training used batches of size 128, step size $T$3 s, and Adam optimizer with a learning rate of $T$4.

Results on the cart-pole system demonstrated that PINODE trajectories closely matched the observed states over both short (2 s) and moderate (0.6 s) rollouts, substantially reducing mean absolute error (e.g., cart position error halved to $T$5 m; pendulum angle error halved to $T$6 rad) compared to a pure ODE model with hand-tuned friction terms. A plausible implication is that PINODE enables data-efficient learning without sacrificing physical plausibility or interpretability—black-box neural ODEs, in contrast, were found to require much larger datasets to converge.

6. Interpretability, Data Efficiency, and Practical Relevance

PINODE maintains a transparent structure: the analytically modeled terms remain explicit, and only the difficult-to-measure residuals are learned. This gives rise to improved interpretability, facilitating model analysis and domain understanding. Furthermore, by leveraging prior physical knowledge and constraining the learning problem, PINODE achieves data efficiency: accurate models can be trained with small datasets (a few hundred swings), compared to black-box approaches that fail to generalize in such settings.

Applications extend to digital-twin simulation, system identification, and model-based control for mechanical systems characterized by partial physical knowledge and challenging non-conservative effects. The PINODE framework offers a principled way to merge first-principles modeling and deep learning, producing forward models with both predictive fidelity and interpretability, as demonstrated in cart-pole experiments (Roehrl et al., 2020).