DG-PINN: Differential Game Based Physics-Informed Neural Network for Vehicle Trajectory Prediction

This paper introduces the Differential-Game based Physics-Informed Neural Network (DG-PINN), a trajectory-prediction model that embeds a multi-agent differential-game formulation into a physics-informed neural network. By treating each vehicle as a strategic agent, DG-PINN simultaneously captures inter-vehicle interactions and individual utilities, thereby unifying longitudinal car-following and lateral lane-changing behaviors within a single, interpretable framework. Physical laws derived from the Hamilton-Jacobi-Bellman equations are incorporated as loss regularizers, enabling the model to retain the data-efficiency and interpretability of physics-based methods while leveraging the representational power of deep learning. We evaluate DG-PINN on the NGSIM I-80 highway dataset under varying training-sample regimes (20-100 sequences) and compare it with constant-velocity/acceleration baselines, a Physics Uninformed Neural Network (PUNN), and an IDM-based PINN. DG-PINN achieves the lowest root-mean-square error, average displacement error, and final displacement error across all sample sizes, outperforming IDM-PINN and PUNN by up to 19 % in low-data settings while exhibiting markedly lower variance. Moreover, the learned utility parameters allow quantitative profiling of driver traits—such as safety-mindedness and aggressiveness—and visualize the rationality of observed versus hypothetical maneuvers. The results demonstrate that integrating differential-game theory with PINNs not only improves prediction accuracy and robustness but also yields actionable insights for explainable decision-making in autonomous-vehicle systems.

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1. DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction The University of Tokyo, Japan Fumihito Furuhashi furuhashi@bin.t.u-tokyo.ac.jp Paper ID: 975 21st Nov. 2025 28th IEEE ITSC @ Gold Coast, Australia
2. / 15 Trajectory Prediction 2 is a key technology for autonomous driving... Directly related to SAFETY • Vehicles dynamically influence one another on the road • Trajectory prediction models anticipate the future paths of surrounding vehicles • Collision-free maneuvers are generated in real time DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Fumihito Furuhashi
3. / 15 Q. What is a Good model？ 3 Answer 1. A High-Accuracy model • Model Size • Sample efficiency (This Work) • Robustness • Data quality • etc. Accuracy NN Transformer GNN Mamba LSTM What shapes prediction Accuracy? DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Fumihito Furuhashi
4. / 15 Q. What is a Good model？ 4 Limitation of Prior Work • Mostly qualitative reasoning • Lacks quantitative validation • Hard to generalize Interpretability: Can we explain why the model made that prediction? Goal-based (Gan et al., 2025) Interaction aware (Gao et al., 2025) Accuracy Interpretability NN Transformer GNN Mamba LSTM GHR IDM Gap Acceptance MOBIL Trade off DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Fumihito Furuhashi Answer 2. A Highly Interpretable model
5. / 15 A Promising Approach: PINN 5 PINN (Physics Informed Neural Networks): • Integrate deep learning with physical models • Enforce theory-based constraints through the loss function Proposed framework Differential Game based PINN But it fails to describe two-dimensional driver motion behavior... ℒ = ℒ𝑑𝑎𝑡𝑎 + ℒ𝑝ℎ𝑦 Physics Law Physics Informed Data Driven 𝑥 𝑣 𝑎 ො 𝑎 DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Fumihito Furuhashi Q. How to model driving behavior on highway? • Car following • Lane-Changing
6. / 15 Position of this Study 6 DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Fumihito Furuhashi Challenge 1 • Trade-off between Accuracy and Interpretability Goal: Propose a socially acceptable trajectory prediction model. Challenge 2 • The decision-making process underlying driving behavior has not been quantitatively interpreted. • The absence of an established theory capable of modeling two-dimensional driving behavior on highways. Proposed framework: Differential Game based PINN
7. / 15 Concept: Differential Game 7 Γ𝑧0 𝑇 ≔ 𝑁, 𝑢𝑖 , 𝐽𝑖 𝑇 , 𝑓 Differential Game: A type of dynamic game with a finite time horizon and a finite number of players. Players Control (actions) Cost function Dynamics State Dynamics Optimal Control DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Fumihito Furuhashi
8. / 15 Concept: Differential Game 8 State Dynamics Optimal Control Total Cost function: 𝐽𝑖 𝑡, 𝑧, 𝑢 = න 𝑡 𝑇 𝑒−𝛾 𝑠−𝑡 𝐿𝑖 (𝑠, 𝑧, 𝑢)𝑑𝑠 𝑢∗ = arg min 𝑢∈𝑈 𝐽 𝑡, 𝑧, 𝑢 Stage cost function 𝑢∗ State Equation: 𝑑𝑧 𝑑𝑡 = 𝑓 𝑧, 𝑢 = 𝑣𝑥 , 𝑣𝑦 , 𝑎𝑥 , 𝑎𝑦 State: 𝑧 = 𝑥, 𝑦, 𝑣𝑥 , 𝑣𝑦 , Control: 𝑢 = 𝑎𝑥 , 𝑎𝑦 DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Fumihito Furuhashi ※Solve HJB eq.
9. / 15 DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Differential Game in this work 9 Assumption Total Cost function: 𝐽𝑖 𝑡, 𝑧, 𝑢 = න 𝑡 𝑇 𝑒−𝛾 𝑠−𝑡 𝐿𝑖 (𝑠, 𝑧, 𝑢)𝑑𝑠 𝑢∗ = arg min 𝑢∈𝑈 𝐽 𝑡, 𝑧, 𝑢 Stage cost function: 𝐿 𝑡, 𝑧, 𝑢 = 1 2 𝑎𝑥 2 + 1 2 𝜃1 𝑎𝑦 2 + 𝜃2 𝑣𝑥 − 𝑣𝑥 𝑑 2 + 𝜃3 𝑣𝑦 − 𝑣𝑦 𝑑 2 +𝜃4 1 + cos 2𝜋 𝑦 𝑙 + ෍ 𝑗≠𝑖 (𝜃5𝑒 − 𝑥𝑖− 𝑥𝑗 2 2𝜎𝑥 2 + 𝜃6𝑒 − 𝑦𝑖− 𝑦𝑗 2 2𝜎𝑦 2 ) Driving style parameter: 𝛽 = 𝑣𝑥 𝑑, 𝑣𝑦 𝑑, 𝜎𝑥 , 𝜎𝑦 , 𝛾 Desired vel. Safety sensitivity Discount rate Fumihito Furuhashi
10. / 15 Differential Game based approach 10 Point 1: Driving Style Params. Point 2: HJB eq. Point 3: State eq. DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Fumihito Furuhashi
11. / 15 Settings 11 https://data.transportation.gov/stories/s/Next-Generation- Simulation-NGSIM-Open-Data/i5zb-xe34/ Dataset Benchmark Metrics NGSIM i-80 dataset • Neural Networks • IDM-PINN • DG-PINN (Proposed) RMSE = 1 𝑁 ෍ 𝑘 𝑁 1 𝑇 ෍ 𝑡 𝑇 𝑥 − ො 𝑥 2 + 𝑦 − ො 𝑦 2 50 folds cross validation. DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Fumihito Furuhashi
12. / 15 Numerical Experiments 12 Good Bad The highest performance and robustness. DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Fumihito Furuhashi • Neural Networks • IDM-PINN • DG-PINN (Proposed)
13. / 15 Cost-Based Evaluation of Predicted Trajectories 13 0.0025 0.20 It selects the trajectory with the minimum cost. Trajectory Cost function value Model Predicted Dummy (Left LC) DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Fumihito Furuhashi
14. / 15 DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Driving style parameter 14 Stage cost function: 𝐿 𝑡, 𝑧, 𝑢 = … + 𝜃2 𝑣𝑥 − 𝑣𝑥 𝑑 2 + … + ෍ 𝑗≠𝑖 (𝜃5𝑒 − 𝑥𝑖− 𝑥𝑗 2 2𝜎𝑥 2 + 𝜃6𝑒 − 𝑦𝑖− 𝑦𝑗 2 2𝜎𝑦 2 ) Desired vel. Safety sensitivity Safety sensitivity = Density Desired velocity Reproduction of the Fundamental Diagram → Evidence of theoretically coherent learning High Density Low Density Fumihito Furuhashi
15. / 15 Concluding Remarks 15 novel Trajectory Prediction framework: Differential Game based-PINN Qualitative side Quantitative side What‘s next? • Achieves the lowest RMSE across all sample sizes. • Up to 19% improvement over NN / IDM-PINN in low-data settings. • High data efficiency and low variance • Learns driving style parameters • Visualizes behavioral rationality through instantaneous cost • Reproduction of the Fundamental Diagram • Adaptation to complex scenarios such as intersections • Overcoming the training complexity of PINNs DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction Fumihito Furuhashi
16. DG-PINN: Differential Game based Physics-Informed Neural Network for Vehicle Trajectory Prediction The University of Tokyo, Japan Fumihito Furuhashi furuhashi@bin.t.u-tokyo.ac.jp Paper ID: 975 21st Nov. 2025 28th IEEE ITSC @ Gold Coast, Australia