Differentiable Nash Planning with Adaptive Probabilistic Safety Certificates Under Interactive Uncertainty

Abstract

Autonomous vehicles must anticipate surrounding motion and how other agents may respond to the ego vehicle. Most stacks still treat prediction, planning, control, and safety as loosely coupled stages. We introduce GTIU (Guided Trajectory Interaction Uncertainty), a differentiable framework that couples a Level-k Interaction-Conditioned Predictor (LkNP), a differentiable weight-adaptive Model Predictive Controller (Diff-MPC), and an interaction-conditioned predictive Control Barrier Function (Pred-CBF; legacy name Game-CBF). Interaction forecasts jointly shape the MPC objective and a predicted-state safety margin, enabling end-to-end learning from a downstream task loss. On recorded driving data and closed-loop simulation, the predictive barrier is less conservative than a fixed-set robust CBF on identical windows, and jointly adapting MPC weights with the safety margin improves tracking. Lane-compliance and collision-free rates are saturated at 100% on non-aggressive agents, so we emphasise minimum gap, tracking error, and clearance margins as primary closed-loop evidence. Code and evaluation scripts are publicly available.

Key Contributions

Closed-Loop Integration

We propagate Level-k forecasts into both the Diff-MPC objective and a predicted-state safety margin, so the same interaction information jointly shapes tracking preferences and clearance.

Differentiable Stack

LkNP and the weight-adaptive Diff-MPC are jointly trained. A parameter-free, differentiable Pred-CBF lets tracking and safety losses propagate through KKT sensitivities to both learned modules.

Adaptive Safety-Margin Control

ASMC jointly selects MPC cost weights and an agent-aware runtime margin scaling for Pred-CBF, with a safe floor that never relaxes the margin below the fixed Pred-CBF baseline.

Expanded Validation

nuScenes mini-val prediction, eight urban closed-loop scenarios, a 40-scenario IDM expansion, a strategic L0/L1/L2 matrix, robustness studies, and a five-seed Interaction-RL ablation.

Methods

Level-k Interaction-Conditioned Predictor (LkNP)

A shared transformer encodes agent histories, cross-attends to the ego Level-1 forecast to form a Level-2 representation, then outputs multimodal trajectories and a policy-type posterior over {L0, L1, L2}. The Level-k recursion is an inductive bias amortised by attention.

LkNP Code

Differentiable Weight-Adaptive MPC (Diff-MPC)

A learned policy π_θ maps interaction forecasts to MPC cost weights. The NMPC solution is differentiable with respect to those weights via implicit differentiation (KKT / IFT), so a downstream task loss can train the weight policy end-to-end.

Diff-MPC Code

Predictive Control Barrier Function (Pred-CBF)

Enforces a classical CBF inequality of the ego dynamics relative to frozen Level-2 predicted positions for that solve. It is a prediction-conditioned barrier, not a fully coupled ego–agent game CBF. No trainable parameters; gradients still flow through it during training.

Pred-CBF Code

Adaptive Safety-Margin Control (ASMC)

Dual-adaptation policy that jointly tunes MPC weights and an agent-aware runtime margin scaling, never relaxing below the fixed Pred-CBF floor. On eight urban scenarios this reduces lateral RMSE 0.134 → 0.126 m and velocity RMSE 5.39 → 4.38 m/s.

ASMC Code

Five-stage closed-loop GTIU pipeline — Figure 2. Closed-loop cycle: observe → Level-k predict → Diff-NMPC → Pred-CBF safety filter → plant. Interaction forecasts jointly shape the MPC objective and the predicted-state safety margin.

Closed-Loop Experiments

Bird’s-eye rollouts of the full stack: observe → Level-k predict → Diff-NMPC → Pred-CBF → plant. GTIU stays in-lane and collision-free, with a positive clearance gap on every catalog scene. Click a plot or storyboard to enlarge.

BEV highlight reel

Closed-loop GTIU-nuScenes · ego green · Pred-CBF clearance overlay

Collision-free Lane 100% ~16 ms / cycle

BEV snapshot: t = 8.9 s · 40 km/h · gap 11.5 m · in-lane · Game-CBF margin 1.4 m.

Scenario 01

Safe overtake

Ego (green) passes a slower lead while Pred-CBF keeps a certified clearance. Predictions, NMPC plan, and the safety filter run every cycle.

Collision-free Lane 100% minADE 0.91 m Gap 11.5 m 16.8 ms / cycle Success 100%

Safe overtake pipeline frames — Figure 3. Five-stage storyboard of the safe overtake. Frame 4 is the Pred-CBF correction.

BEV snapshot: ~40 km/h · gap 13.6 m · in-lane · Pred-CBF blocks an unsafe close-in.

Scenario 02

Slow-lead following

When a lead vehicle slows, Diff-NMPC adapts cost weights from Level-k intent and Pred-CBF blocks an unsafe close-in. The ego stays in-lane with a 13.6 m gap.

Collision-free Lane 100% Gap 13.6 m 16.8 ms / cycle Success 100%

Slow-lead closed-loop sequence — Figure 4. Slow-lead sequence: prediction → planning → Pred-CBF correction → executed control.

Closest point: gap 2.1 m. The barrier fires (red box), the ego yields, then recovers.

Scenario 03

Yield / avoid

On a tight merge the barrier tightens (gap 2.1 m at the closest point). GTIU yields rather than colliding, then recovers. Collision-free rate remains 100% on this catalog.

Collision-free Closest gap 2.1 m Success 100% 16.8 ms

Yield and avoid closed-loop sequence — Figure 5. Yield/avoid: Pred-CBF fires and the ego opens a safe gap.

Safe cross: gap 18.4 m · 29 km/h · Pred-CBF margin 0.00 m at the decision instant, then recovery.

Scenario 04

Pedestrian crosswalk

The ego yields to a crossing pedestrian. Level-k intent plus Pred-CBF keep a large gap (18.4 m) while remaining in-lane at ~29 km/h.

Collision-free Lane 100% Gap 18.4 m minADE 0.91 m Success 100%

Pedestrian crosswalk pipeline — Figure 6. Pedestrian safe-cross storyboard. Frame 4 is the Pred-CBF check at the crosswalk.

Closed-loop urban BEV grid of eight scenarios — Figure 7. All eight urban closed-loop scenes (BEV): highway merge, unprotected left, aggressive cut-in, roundabout, pedestrian, multi-lane highway, four-way intersection, urban corridor.

Real nuScenes Boston-seaport camera and BEV overlays — Figure 8. Real nuScenes Boston-seaport windows: camera context with prediction / avoidance and ground-truth plus ego plan. Yield, overtake, and safe-cross labels are the executed GTIU decisions.

Result Plots

Headline comparisons from the GTIU experiments. Click any plot to enlarge. Mini-val is not a full nuScenes-val SOTA claim; lane and collision rates are saturated, so we emphasise minADE, margin, gap, and tracking error.

Headline GTIU result cards — Headline numbers: minADE 0.71 m · Game-CBF margin +0.96 m · lat. RMSE 0.126 m · 15.8 ms · CommonRoad 100%.

Experimental story verifying each claimed contribution — Each claimed contribution is measured: prediction, planning, safety, runtime, and leave-one-out ablation.

minADE vs SOTA — GTIU-Roadmap minADE 0.71 m vs MTR 0.87 m, Wayformer 1.33 m, CoverNet 2.00 m (mini-val / published numbers; splits differ for MTR).

Mode selection ADE — Deployable GTIU-LR ADE 1.52 m vs naive weight-argmax 3.29 m — close to the 1.50 m oracle.

Simulation vs latest papers — Four-way strip: prediction, urban closed-loop, CommonRoad success (100%), and 16.8 ms runtime.

Master benchmark Interaction RL — ASMC (Interaction RL v2) is safety-certified, 100% collision-free, and lowest urban lateral RMSE (0.126 m).

Open vs closed loop summary — Open-loop ADE, closed-loop urban tracking, and Monza-style NMPC comparison vs Diff-WMPC.

Capability win matrix — Capability matrix across prediction, safety, Nash, Pred-CBF, E2E, urban closed-loop, CommonRoad, and ≤20 ms.

Safety margin vs baselines — Mean safety margin +0.96 m for full GTIU vs −2.93 m for a robust CBF on the same windows.

CommonRoad success: GTIU 100% vs Moller RL 81% and B125 0–58%.

Adaptive margin per scenario — Per-scenario CBF margin: ASMC never drops below the fixed Pred-CBF floor.

Uncertainty bars — Safety margin, tracking spread, and runtime 16.8 ± 1.1 ms (well under a 50 ms budget).

Leave-one-out ablation — Removing Pred-CBF collapses the safety margin to +0.07 m. Full GTIU keeps +0.96 m at 0.91 m minADE.

Module training ablation — Joint predictor + NMPC training is what moves open-loop minADE; closed-loop lateral error stays ~0.12–0.13 m.

Runtime breakdown — 16.8 ms cycle: Diff-NMPC 6.5 ms, LkNP 6.2 ms, Pred-CBF 2.8 ms (~60 Hz).

Stress and failure analysis — Honest stress map: aggressive cut-in is the tracking outlier (0.51 m); CBF stays non-negative on most windows.

Quantitative Results and Benchmarking

The overall best closed-loop performance is achieved by the full GTIU stack with ASMC, running at 15.8 ms per cycle (≈ 63 Hz) on the reported hardware.

0.71 m

minADE₆ (nuScenes mini-val)

Lateral RMSE reduction (ASMC)

19%

Velocity RMSE reduction (ASMC)

15.8 ms

Full-stack cycle time (≈ 63 Hz)

Open-loop path prediction on nuScenes mini-val

N = 256 agent windows, 10 mini scenes. Lower is better. GTIU attains the lowest same-split minADE₆ and minFDE₆.

Method	minADE₆ ↓	minFDE₆ ↓	ADE ↓	MR@2 m ↓
MTP-style anchor	1.18	2.33	1.39	0.35
Const. speed/yaw	1.39	2.73	1.39	0.96
GTIU-MOCL (baseline)	0.91	2.60	3.29	0.98
GTIU-Roadmap (ours, oracle)	0.71	1.29	2.50	0.25
GTIU-LR (ours, deployable)	0.71	1.29	1.52	0.98

ADE is not the same as minADE₆. Roadmap ADE (2.50 m) is mixture / primary-mode ADE; minADE₆ is hindsight best-of-6. GTIU-LR ADE (1.52 m) is the deployable top-1 prediction handed to the controller. Mini-val is not a full nuScenes-val SOTA claim.

Closed-loop self-ablation on 8 urban scenarios

Lane compliance and collision-free rate are saturated at 100% for all variants, so they are secondary. ASMC reduces lateral RMSE by 6% and velocity RMSE by 19% versus fixed Pred-CBF.

Variant	Lat. RMSE [m]	Vel. RMSE [m/s]	Lane [%]	Coll. [%]	Margin [m]
Pred-CBF (fixed)	0.134	5.39	100	100	27.97
Agent-Aware Pred-CBF	0.134	5.39	100	100	28.25
ASMC (ours, best)	0.126	4.38	100	100	28.05

Secondary closed-loop safety metrics

Worst-case minimum gap is 2.15 m on this set. Minimum gap and CBF margin are the discriminating safety metrics.

Variant	Lat. RMSE	Vel. RMSE	Worst min gap [m]	min h_CBF
Fixed Pred-CBF	0.134	5.39	2.15	8.78
Agent-aware Pred-CBF	0.134	5.39	2.15	8.67
ASMC (GTIU)	0.126	4.38	2.15	8.76

Per-scenario closed-loop metrics (full GTIU stack)

Scenario	min gap [m]	Mean margin [m]	Lat. RMSE	Vel. RMSE
highway_merge	2.15	47.3	0.040	3.70
unprotected_left	2.15	11.6	0.018	5.18
aggressive_cut_in	2.15	27.3	0.509	7.49
roundabout_entry	8.49	11.5	0.068	2.45
pedestrian_crosswalk	19.06	22.8	0.001	2.01
multi_lane_highway	2.87	62.8	0.206	6.30
four_way_intersection	4.60	8.3	0.015	5.08
urban_corridor_4lane	2.17	33.3	0.215	2.35

Aggressive cut-in is the tracking outlier (0.509 m lateral RMSE). Several scenarios share the 2.15 m floor on minimum gap.

Inference latency

Method	Latency [ms] ↓	Rate [Hz] ↑
GTIU (ours, full stack)	15.8 ± 0.22	63
MTR	45.0	22
Diff-WMPC	8.0	125

GTIU: mean ± std over 10⁴ cycles (batch size 1, i7-12700K, RTX 3080, PyTorch 2.1 / CUDA 12.1); 95th percentile 16.1 ms. Hardware-specific; not a real-time certification.

BibTeX

@misc{Singh2026GTIU,
  author={Singh, Anand and Betz, Johannes and Piccinini, Mattia},
  title={Differentiable Nash Planning with Adaptive Probabilistic Safety Certificates Under Interactive Uncertainty},
  year={2026},
  note={manuscript in preparation}
}