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A Physics-Informed Deep Learning Paradigm for Car-Following Models

Zhaobin Mo, Xuan Di, Rongye Shi

arXiv:2012.13376v4cs.LGeess.SP

TL;DR

Physics-based car-following models are interpretable and data-efficient but simplify driving cognition, whereas deep learning models are generalizable yet data-hungry and less interpretable. The paper develops PIDL-CF architectures that encode IDM and OVM in neural-network models, support prediction and parameter discovery, and combine observed with physics-derived training information. Across simulations and NGSIM, PIDL outperforms its individual components, particularly with sparse measurements, while the framework remains subject to unresolved questions about richer models, model selection, and driver heterogeneity.

  • Problem

    Physics-based car-following models may not capture complex cognition, while deep learning models require much data and lack interpretability.

  • Method

    The paper builds PIDL-CF neural architectures encoding IDM and OVM, with observed and collocation data supporting acceleration prediction or joint parameter discovery.

  • Results

    PIDL outperforms pure physics-based and data-driven models across varied training-data sizes, especially when measurements are sparse.

  • Takeaways & Limitations

    PIDL-CF provides a framework for data-efficient, generalizable car-following modeling and potential system identification of driving models.

  • Takeaways & Limitations

    The framework leaves unresolved the encoding of more advanced CFMs, systematic physics-model identification, and driver heterogeneity.

Abstract

from arXiv · show

Car-following behavior has been extensively studied using physics-based models, such as the Intelligent Driver Model. These models successfully interpret traffic phenomena observed in the real-world but may not fully capture the complex cognitive process of driving. Deep learning models, on the other hand, have demonstrated their power in capturing observed traffic phenomena but require a large amount of driving data to train. This paper aims to develop a family of neural network based car-following models that are informed by physics-based models, which leverage the advantage of both physics-based (being data-efficient and interpretable) and deep learning based (being generalizable) models. We design physics-informed deep learning car-following (PIDL-CF) architectures encoded with two popular physics-based models - IDM and OVM, on which acceleration is predicted for four traffic regimes: acceleration, deceleration, cruising, and emergency braking. Two types of PIDL-CFM problems are studied, one to predict acceleration only and the other to jointly predict acceleration and discover model parameters. We also demonstrate the superior performance of PIDL with the Next Generation SIMulation (NGSIM) dataset over baselines, especially when the training data is sparse. The results demonstrate the superior performance of neural networks informed by physics over those without. The developed PIDL-CF framework holds the potential for system identification of driving models and for the development of driving-based controls for automated vehicles.

1. Motivation

Car-following models balance interpretable physics-based formulations against generalizable but data-hungry and less interpretable deep learning approaches. The paper proposes physics-informed architectures that combine these strengths across models, regimes, and training settings.

  • Modeling gap: Physics-based models use predefined mathematical forms and few parameters, but simplify cognition and may generalize poorly across diverse interactive driving scenarios.
  • Modeling gap: Deep learning models learn from observations without a predefined form and can capture asymmetric driving behavior, but require substantial data and offer limited interpretability.
  • Proposed direction: PIDL combines a neural network with encoded physics to address the complementary limitations of physics-based and data-driven car-following models.
  • Contributions: The paper designs PIDL architectures using IDM and OVM to predict acceleration in acceleration, deceleration, cruising, and emergency-braking regimes.
  • Contributions: PIDL also supports joint acceleration prediction and model-parameter discovery, rather than requiring physics parameters to be calibrated before neural-network training.
  • Contributions: Experiments with numerical simulations and NGSIM compare PIDL against pure physics-based and data-driven models, including ANN and LSTM variants.

3. PIDL for Identification of Car-Following Behavior

PIDL-CF uses observed states and unobserved collocation states to train a neural network alongside a physics-based computational graph. It supports prediction-only training with calibrated physics and joint estimation that learns physics parameters.

  • Problem formulation: PIDL-CF defines observed states as spacing, velocity difference, and subject-vehicle velocity, with acceleration as the observed action.
  • PIDL-CF problems: Prediction-only training uses a known physics-based model whose parameters are calibrated before the PIDL model is trained.
  • PIDL-CF problems: Joint estimation predicts acceleration while simultaneously estimating physics parameters within a feasible physical domain.
  • Architecture: The framework contains a physics-uninformed neural network mapping observed states to acceleration and a physics-based graph mapping collocation states to physics-based acceleration.
  • Architecture: The architecture can use ANN-based PUNNs and may also adopt recurrent neural networks; the paper uses tanh activation for its ANN implementation.
  • Data construction: Collocation data are generated by evaluating physics-based accelerations at selected states, while observed data are split into training, validation, and test subsets.

Calibration of the Physics

The prediction-only problem first calibrates the physics-based car-following model against observed acceleration data, using an optimization objective that measures mean-squared discrepancy.

  • Physics calibration estimates the parameters of a traditional car-following model from observed training data.The paper describes calibration as parameter estimation for a physics-based CFM.
  • The objective function measures the mean-squared discrepancy between physics-estimated and ground-truth acceleration.The paper distinguishes this calibration objective from the later neural-network training loss.
  • Calibration methods include gradient-based approaches such as least squares, maximum likelihood, and sequential quadratic programming, alongside gradient-free genetic algorithms.

Training of PUNN

PIDL-CF trains a neural network with physics-derived acceleration information, either after calibrating physics beforehand or while jointly estimating physics parameters and network weights.

  • In prediction-only training, calibrated physics parameters are fixed while the PUNN parameters are optimized using observed and collocation data.The calibrated physics supplies predicted accelerations for the physics-informed training process.
  • The PUNN is trained with Xavier initialization, Adam optimization, validation MSE, and early stopping to recover parameters with minimum validation error.The selected model is evaluated on test data after training stops.
  • In joint estimation, the PUNN and physics parameters are trained simultaneously, so the loss and physics predictions vary with λ.The physics discrepancy is trainable with respect to the physics parameters, whereas the observed-data discrepancy is unrelated to λ.
  • Joint training uses separate learning rates for the PUNN and physics model and clips physics gradients to help respect physical parameter bounds.The paper motivates these choices by differences in parameter scales, gradients, and convergence behavior.
  • After training, the learned PUNN receives current states and outputs acceleration at the next timestep, with test MSE used for evaluation.The joint-estimation variant also returns learned physics parameters λ∗.
  • The generic PIDL-CF framework contains prediction-only and joint-estimation formulations, differing in which parameters are updated and which states and actions support those updates.Figure 4 presents the corresponding computational flows, while Table 2 summarizes their parameter-update differences.
  • The prediction-only loss combines observed-data discrepancy with discrepancy between PUNN and physics predictions on collocation states.These components are denoted MSEO and MSEC in the generic architecture.

4. PIDL for Existing Physics-based CFMs

The paper evaluates PIDL-CF by encoding established physics-based car-following models, IDM and OVM, and testing both prediction-only and joint-estimation problems on simulated trajectories with artificial noise.

  • The simulated-data experiments use trajectories generated from known physics-based models, enabling direct comparison with ground-truth mechanisms and parameters.The controlled simulations provide a stepping stone before evaluation on real-world noisy data.
  • Both PIDL-IDM and PIDL-OVM solve prediction-only and joint-estimation problems using noisy trajectories generated from IDM and OVM.Prediction performance is compared with ground truth and physics-uninformed neural networks.
  • Experiment settings: The simulated experiments specify model and training settings in Table 3 and separately detail data generation, collocation data, physics calibration, and sensitivity analysis.
  • PIDL-IDM architecture: IDM parameters used to generate ground-truth trajectories include desired velocity, desired time headway, minimum spacing, maximum acceleration, comfortable deceleration, and a constant δ.The ground-truth parameter values are listed in Table 3.
  • PIDL-IDM architecture: The PIDL-IDM architecture combines a PUNN with an IDM-encoded physics computational graph whose parameters can be pre-calibrated or jointly trained, except δ, usually fixed at 4.
  • PIDL-OVM architecture: The PIDL-OVM architecture similarly combines a PUNN with an OVM computational graph parameterized by vmax, hc, and k.The OVM graph uses the hyperbolic tangent operator in its desired-velocity formulation.

Generation of training data

For the PIDL-IDM prediction-only experiment, trajectories are generated separately for accelerating, decelerating, cruising, and emergency-braking regimes, with regime-specific sampling and data splits.

  • The four regimes are accelerating, decelerating, cruising, and emergency braking, defined by the subject vehicle’s acceleration over the simulation.Cruising lasts 20 seconds; other regimes end when acceleration reaches zero.
  • Each regime uses one PIDL-IDM model trained and tested on trajectories from that same regime.
  • Initial subject spacing and velocity are varied while preserving the acceleration regime, and the leader’s initial velocity is varied before remaining constant within each run.
  • Sample trajectories display time-space position, speed, and acceleration profiles by regime, with Gaussian noise added to the ground-truth trajectories.
  • Each regime is split into 50% training, 25% validation, and 25% testing data, while the number of observed training points is varied.

Generation of collocation data

Collocation data are generated by sampling states from the state space and computing their accelerations with IDM.

  • Each experiment uses 20 collocation data points, within the range of the varying training-data sizes.
  • The collocation states are randomly selected from the state space, and IDM computes their acceleration values.States and computed accelerations together form the collocation data.

Sensitivity analysis

The prediction-only experiments vary the loss-function weight α, while the joint-estimation experiments fix α at 0.7 to focus on simultaneous prediction and parameter discovery.

  • Prediction-only problem: Prediction-only sensitivity analysis varies α and identifies α∗ as the value producing the minimum test error.
  • Prediction-only problem: When α = 1, only the data-discrepancy term remains, reducing PIDL-IDM to a pure PUNN model.
  • Joint-estimation problem: Joint-estimation experiments fix α = 0.7 while varying nO to assess the effect of training-data quantity.The fixed weight isolates simultaneous acceleration prediction and parameter discovery because α’s effect was studied separately.

Parameters tuning

The experiments tune neural-network and optimization settings, then evaluate PIDL-IDM across traffic regimes and problem types using PUNN as the principal baseline.

  • Parameters tuning: Tuned parameters include collocation-data count, learning rates for PUNN and physics, and PUNN hidden-layer architecture.
  • Parameters tuning: The search considers 1–5 hidden layers and 30–512 neurons per layer before selecting settings through comparative experiments.
  • PIDL-IDM experiments: In prediction-only experiments, ground-truth IDM parameters are known; joint estimation trains physics parameters together with the PUNN.
  • PIDL-IDM prediction-only results: At epoch 53, PIDL-IDM reaches minimum validation loss for the accelerating regime with (nO, nC, α) = (20, 20, 0.1).The PUNN parameters at that epoch are used as the optimal parameters θ∗.
  • PIDL-IDM prediction-only results: PIDL-IDM outperforms PUNN in all four regimes, with MSE reduction ranging from 1.26% in cruising to 16.80% in emergency braking.The advantage is strongest when training data are smaller.
  • PIDL-IDM prediction-only results: The optimal α∗ is lower with small training sets and approaches one as training data increase, shifting emphasis from physics discrepancy toward data discrepancy.
  • PIDL-IDM joint-estimation results: With 180 or more training data points, PIDL-IDM achieves satisfactory acceleration prediction and IDM-parameter estimation; gains diminish beyond 260 observations.
  • PIDL-OVM experiments: PIDL-OVM combines a PUNN with an OVM computational graph, whose parameters include vmax, hc, and k.

Generation of training data

The real-world evaluation uses processed and behavior-matched NGSIM car-following cases to test PIDL-IDM and PIDL-OVM under unknown trajectory-generating physics.

  • Simulated training data: OVM trajectories are simulated over 20 seconds with Gaussian noise, a 0.1-second timestep, and 84k total training data.
  • PIDL-OVM evaluation: PIDL-OVM preserves its prediction advantage over PUNN when the encoded physics changes from IDM to OVM.
  • PIDL-OVM joint estimation: With 180 or more training data points, PIDL-OVM achieves satisfactory acceleration prediction and OVM-parameter estimation.
  • NGSIM evaluation: NGSIM evaluation judges PIDL models by predicted accelerations because the mechanism generating real-world trajectories is unknown.
  • NGSIM data: NGSIM provides high-resolution vehicle information every 0.1 seconds from a US Highway 101 camera dataset focused on automobile trajectories.
  • Pre-processing: Speed and acceleration are smoothed using median-based processing and Savitzky–Golay filters to reduce high-frequency and remaining noise.
  • Car-following case selection: A car-following case requires same-lane vehicles, spacing no greater than 150 meters, and duration longer than 10 seconds.
  • Behavior matching: Behavioral heterogeneity is reduced by normalizing eight calibrated IDM and OVM parameters, selecting 50 cases nearest the feature-space centroid, and retaining 45K data points.

PIDL models

The study compares PIDL car-following models built with IDM and OVM physics components against neural-network and physics-based baselines. It evaluates prediction-only and joint-estimation formulations using one-step and trajectorial calibration.

  • PIDL-CF evaluation uses IDM and OVM as encoded physics components, with shared PUNN architectures for prediction-only and joint-estimation problems.
  • Baseline comparisons include a standalone PUNN and physics-based IDM and OVM models.
  • Physics baselines use one-step least-squares calibration or trajectorial genetic-algorithm calibration, respectively minimizing instantaneous acceleration error or whole-trajectory error.

Evaluation methods and metrics

The evaluation measures acceleration, position, and velocity errors across training-data sizes and compares PIDL variants with neural, physics-based, and alternative-architecture baselines. Results consistently favor physics-informed models, particularly under limited data, while joint estimation and model complexity affect performance.

  • One-step comparison: PIDL-IDM and PIDL-OVM outperform the PUNN, especially with insufficient training data, while standalone IDM and OVM perform worst.
  • One-step comparison: PIDL-IDM outperforms PIDL-OVM in both formulations, consistent with IDM providing more accurate physics information and receiving a higher physics weight.
  • One-step comparison: Joint-estimation PIDL-IDM has lower test error than prediction-only PIDL-IDM, whereas the two PIDL-OVM formulations have close test errors.
  • One-step comparison: Joint-estimation parameters converge toward realistic ranges while the models jointly predict accelerations and estimate physics parameters on real-world data.
  • Trajectorial comparison: Prediction-only PIDL models outperform baselines in position and velocity errors, with larger advantages under insufficient training data and trajectorial calibration.
  • Physics-based car-following models: Among calibrated physics components, IDM performs best with fewer than 25 training data points, motivating its selection alongside GHR and FVDM for sparse-data PIDL experiments.
  • Physics-based car-following models: All PIDL models outperform the ANN, PIDL-IDM leads below 50 data points, and joint estimation reduces errors for IDM, FVDM, and GHR.
  • Neural network architecture: LSTM-based PIDL-IDM performs best at a 1-second gap but requires more data than ANN-based PIDL, while longer gaps further increase data demands.

6. Conclusions

The paper develops a physics-informed deep-learning framework for car-following that combines physics-based models with neural networks. Experiments support acceleration prediction and physics-parameter estimation, while several extensions remain unresolved.

  • Conclusions: PIDL-CF encodes IDM and OVM into neural-network architectures for car-following modeling.The framework includes a physics-informed loss combining deviation from observed data with distance from physics predictions.
  • Conclusions: PIDL-CF outperforms physics-uninformed neural networks on simulated acceleration-estimation tasks and is also evaluated on real-world NGSIM data.The numerical experiments use data generated from IDM or OVM, followed by tests on NGSIM data.
  • Conclusions: The framework predicts accelerations and estimates physics parameters on real-world data, supporting its use for driving-model system identification and driving-based automated-vehicle controls.These applications are stated as potential benefits of the developed PIDL-CF framework.
  • Conclusions: Future work includes encoding more advanced car-following models, systematically identifying suitable physics models, and accounting for driver heterogeneity.The authors identify these as unresolved research directions.

Appendix A. Metrics for model accuracy

The appendix describes prediction-accuracy metrics and supplementary experiments on metric choice and neural-network structure. It reports similar test performance across training metrics and examines generalization across network architectures.

  • Appendix A. Metrics for model accuracy: MSE measures prediction accuracy, while MAE, RMSE, and RMSPE are also tested for physics calibration.The experiments use MSE in Sections 4 and 5 and try three additional calibration metrics.
  • Appendix A. Metrics for model accuracy: Different training metrics achieve similar performance on the same NGSIM US-101 test dataset, supporting the framework’s reported agnosticism to training metrics.Evaluation uses MSE uniformly, while training-data size is varied as in Sections 4 and 5.
  • Appendix A. Metrics for model accuracy: The architecture study varies hidden-layer count and neurons per layer, using held-out test error to assess generalization in a numerical accelerating-regime experiment.The study uses 100 training observations, α = 1, and early stopping during training.
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