Conditional Invertible Neural Networks for Data-Driven UAV Control: A 2-D Proof of Concept Christian Wittke ∗ Stephan Myschik ∗∗ Oliver Niggemann ∗ ∗
arXiv:2607.13703v1 [cs.LG] 15 Jul 2026
Computer Science in Mechanical Engineering, Helmut Schmidt University, Hamburg, Germany. (e-mail: christian.wittke;[email protected]) ∗∗ Institute for Aeronautical Engineering, University of the Bundeswehr Munich, Bavaria, Germany (e-mail: [email protected]) Abstract: We investigate conditional invertible neural networks (cINNs) as probabilistic inverse-dynamics models for multirotor control. For a planar X8 coaxial multicopter, we learn p(u | st , ct ) from an incremental nonlinear dynamic inversion (INDI) teacher using rationalquadratic spline coupling and invertible linear mixing. Open-loop reproduction reaches R2 = 0.944, mean CRPS 0.0915, and log-probability–error correlation ρ = −0.60. Over 15 closed-loop scenarios, position RMSE matches INDI (9.7 vs. 9.5 m) with 47 % tracking acceptably; failures separate into attitude divergence under aggressive steps and phase lag under high-frequency references, isolating command bandwidth and data coverage as dominant failure mechanisms. Keywords: Unmanned aerial vehicles, Flight control, Learning and adaptive systems, Neural networks, Probabilistic methods 1. INTRODUCTION Multirotor control has matured to the point where timeoptimal trajectories are flown by both model-based and learned controllers, yet a gap remains between the two paradigms. Classical linear control methods such as PID and model predictive control (MPC), as well as controllers based on (incremental) nonlinear dynamic inversion (NDI/INDI), are predictable and certifiable but rely on accurate system models. Data-driven methods generalise from experience but return point estimates without any associated confidence. Neither is sufficient in safetyrelevant deployments, where the operator needs both adaptability to unmodelled effects and a quantitative signal that flags when the controller leaves its trusted regime. This paper treats multirotor control as a probabilistic inverse dynamics problem. Given the current state st and a tracking command ct , we model the full conditional distribution p(u | st , ct ) (1) over the motor command u ∈ Rnu instead of predicting a single value. The mode of p(u | st , ct ) serves as the deterministic control output; its dispersion provides a sample-wise uncertainty estimate that contracts where training data is dense and spreads where it is sparse, the behaviour conditional flows exhibit on other inverse problems (Ardizzone et al., 2019; Denker et al., 2021). The construction mirrors INDI, which builds its control update from the current state and a desired acceleration encoded implicitly in ct through the velocity command and tracking errors. To represent (1) we use a conditional invertible neural network (cINN) built from normalising flows (Dinh et al., © 2026 the authors. This work has been accepted to IFAC for publication under a Creative Commons Licence CC-BY-NC-ND.
2017; Kingma and Dhariwal, 2018; Durkan et al., 2019; Ardizzone et al., 2019). cINNs combine three properties relevant here: a bijection between data and a tractable base distribution, exact likelihood evaluation via the change-ofvariables formula, and efficient sampling. We use rationalquadratic spline coupling (Durkan et al., 2019) for the motor space, activation normalisation (Kingma and Dhariwal, 2018) for stable optimisation, and learned invertible linear mixing for cross-channel coupling, with sine–cosine encoding of attitude angles in the conditioning network. The testbed is a planar X8 coaxial multicopter restricted to the XZ-plane, with eight motor inputs and reference velocity commands up to ±2 m/s in x and z. The simulation framework extends a Simulink model used in prior sparseidentification work on the same vehicle (Tappe et al., 2025; Kelm and Myschik, 2023). An INDI teacher supplies both the training data and the closed-loop baseline. The trained cINN reproduces the teacher across all eight motor channels at R2 = 0.944 and matches the INDI baseline in mean position RMSE (Root Mean Square Error) over 15 closedloop scenarios. Two failure modes emerge—attitude divergence under aggressive single-step commands and phase lag under high-frequency references—both with concrete physical origins. The main contributions are: • a probabilistic inverse-dynamics formulation p(u | st , ct ) that conditions on operationally available signals rather than the unobservable future state st+1 ; • a cINN architecture combining rational-quadratic spline coupling, activation normalisation, and learned invertible linear mixing, with sine–cosine attitude encoding;
• an open- and closed-loop evaluation on a 2-D X8 with an INDI teacher as both data source and baseline, isolating command bandwidth and training-data coverage as the dominant failure mechanisms. Section 2 reviews related work; Section 3 formalises the problem and the cINN architecture; Section 4 the experimental setup; Section 5 the results; Section 6 concludes with future work.
(Ardizzone et al., 2019) model p(x | c) and have been deployed across inverse problems in medical imaging (Denker et al., 2021; Nölke et al., 2021), materials design (Kumar et al., 2021), particle physics (Heredge et al., 2024), industrial vision (Wittke et al., 2025), and continuum-robot inverse kinematics (Rao et al., 2023). The probabilistic inverse-dynamics setting on aerial vehicles has, to our knowledge, not been explored. 3. SOLUTION
2. STATE OF THE ART 2.1 Multirotor Control
3.1 Problem Formulation
Cascaded proportional-integral-derivative (PID) loops remain the dominant control architecture for commercial multirotors thanks to their transparency and wellunderstood tuning procedures (Lopez-Sanchez and MorenoValenzuela, 2023; Chao et al., 2010). Linear control laws, however, struggle with the coupling and nonlinearities of aggressive flight, which has motivated nonlinear extensions: model predictive control (MPC) optimizes over a receding horizon and has enabled near time-optimal racing performance (Romero et al., 2022; Foehn et al., 2021), while incremental nonlinear dynamic inversion (INDI) treats the inverse of the rotational dynamics explicitly and has become the de facto standard for aggressive flight on resource-constrained platforms (Smeur et al., 2016). Both approaches require accurate parameter knowledge and provide no native measure of prediction confidence— a gap that becomes critical outside the trusted operating regime.
The inverse-dynamics problem asks for the motor command u ∈ Rnu that drive the vehicle from st ∈ Rns along an external reference. Conditioning on the desired next state st+1 is unavailable in closed loop, since st+1 is not observable when the command must be issued. We condition instead on the signals an outer loop produces in practice: the velocity command vtcmd and the position and velocity tracking errors epos , evel , collected in cmd t post vel (2) ct = vt ; et ; et ∈ Rnc This is the same information INDI uses to construct its control update. We learn the full conditional density p(u | st , ct ), which captures the teacher’s response together with the dispersion induced by limited training coverage.
2.2 Learning-Based and Imitation Approaches Supervised imitation of expert controllers is a wellestablished paradigm for synthesising fast neural policies (Dierks and Jagannathan, 2009; Loquercio et al., 2020). Reinforcement learning removes the need for an explicit teacher and has produced champion-level drone racing controllers (Kaufmann et al., 2023), with comparable performance limits reported for both RL and optimal control under appropriate conditions (Song et al., 2023). Hybrid strategies combine learned components with structural priors, including physics-informed neural networks (Cheng et al., 2024) and qualitative system models (Tappe et al., 2025). Almost all of these controllers emit deterministic motor commands; where uncertainty is reported, it relies on Monte Carlo dropout or deep ensembles, which lack a principled likelihood interpretation. We position the present work as supervised imitation of an INDI teacher with the network’s probabilistic output replacing such adhoc estimators. 2.3 Conditional Normalizing Flows for Inverse Problems Normalizing flows model complex densities as compositions of invertible transformations, enabling exact likelihood evaluation through the change-of-variables formula (Dinh et al., 2017; Kingma and Dhariwal, 2018). Affine coupling layers and learned 1×1 mixings provide tractable Jacobians, while rational-quadratic spline flows substantially increase expressiveness without sacrificing analytic invertibility (Durkan et al., 2019). Conditional extensions
3.2 System Model The vehicle is an X8 coaxial multicopter with eight individually controlled rotors in four counter-rotating coaxial pairs. The plant runs in Simulink, exposed to Python as a precompiled C library through the RMT-CopterGym environment of Tappe et al. (2025). The motor command is u = [u1 , . . . , u8 ]⊤ ∈ R8 , and the state aggregates position, linear velocity, attitude, and body angular rates: s = [p; v; Θ; ω] ∈ R12 with Θ = [ϕ, θ, ψ]⊤ . For the proof of concept the simulation is restricted to the XZplane: lateral position and velocity, roll, yaw, and the corresponding angular rates remain identically zero. The full 12-D state is retained so the architecture extends to 3-D flight without modification. 3.3 Conditional Invertible Neural Network The cINN learns a parametric bijection fθ ( · ; h) : Rnu → Rnu , z = fθ (u; h), that maps motor commands to a latent vector z ∼ pz = N (0, Inu ). A learned encoder gθ produces the conditioning vector h ∈ Rnh from (st , ct ). The changeof-variables formula gives the conditional density of u in closed form: ∂fθ log p u | st , ct = log pz fθ (u; h) + log det (3) ∂u A standard normal base is sufficient: spline-based bijections are universal density approximators (Durkan et al., 2019). 3.4 Conditioning Encoder Three of the twelve state components are angles (Θ = [ϕ, θ, ψ]⊤ ) and contain a 2π-discontinuity that an MLP cannot fit smoothly. We replace them by their sine–cosine
encoding, yielding a 15-D state and a 24-D encoder input together with the 9-D command. The encoder gθ is a three-layer MLP with hidden dimension 256 and SiLU activations, producing h ∈ R256 which conditions every flow block. 3.5 Flow Architecture The flow fθ stacks K = 9 blocks, each combining three invertible sub-layers (Fig. 1). Activation normalisation (Kingma and Dhariwal, 2018) applies an affine per-channel scale with parameters initialised from the first mini-batch. A rational-quadratic spline coupling layer (Durkan et al., 2019) splits the 8-D input into two halves; one half is transformed componentwise by a monotonic piecewise spline (Ks = 8 bins on [−B, B]) whose parameters are predicted from the other half and h. Spline coupling outperforms affine coupling (Dinh et al., 2017) here because motor allocation in a coaxial X8 is strongly nonlinear. A learned 8×8 linear map (Kingma and Dhariwal, 2018), initialised orthogonal, mixes across channels and compensates for the static coupling split. The full network has about 1.5×106 trainable parameters. 3.6 Training and Inference Training minimises the negative log-likelihood of u over the dataset D, with log p(u | st , ct ) evaluated via (3). At inference, the deterministic control output is the mode ût = fθ−1 (0; h); for uncertainty quantification we draw N = 20 latent samples through the inverse flow and report the empirical mean and componentwise standard deviation. 4. METHODOLOGY 4.1 Simulation and Data All experiments run inside the RMT-CopterGym environment of Tappe et al. (2025) at a fixed control rate of 100 Hz. The simulation is constrained to the XZ-plane and uses the full 12-D simulator state as conditioning input. Training data is generated by an INDI teacher (Smeur et al., 2016); INDI uses the measured (here: simulated) angular acceleration to invert the rotational dynamics incrementally, yielding stable high-bandwidth velocity tracking without requiring a precise inertia model. The same INDI controller also serves as the closed-loop baseline. Reference signals are piecewise-constant velocity commands vtcmd ∈ R3 with non-zero entries only in x and z, step amplitudes drawn uniformly from ±2 m/s, switching times from a coarse 1–3 s grid. Each scenario runs for 12.5 s (1 252 samples); the full dataset comprises approximately 1000 scenarios totalling ∼1.25×106 tuples (st , ct , ut ), with ct constructed online from the instantaneous tracking errors. States, commands, and controls are standardised channelwise using training-split statistics, frozen for all subsequent use. Trajectories are split scenario-wise into training, validation, and test sets (75/15/10 %), so every time-step belongs to exactly one split.
4.2 Network and Training Table 1 summarises the architecture and the training hyperparameters. The model is optimised with AdamW on the NLL objective, with a linear warmup over 5 epochs and cosine annealing of the learning rate to 10−6 . An exponential moving average of the weights is used at evaluation. We use early stopping on the validation NLL: training is terminated when the validation loss has not improved for 15 consecutive epochs. With this rule, training stops around epoch 75; total wall-clock time on a single NVIDIA RTX 3090 is 2 h 52 min. Table 1. cINN architecture and training configuration. State / command / control dim Encoder Context dim nh Flow blocks × layers K Spline bins Ks , tail bound B Trainable parameters
12 / 9 / 8 3 × 256, SiLU 256 3×3=9 8, 5.0 ∼1.5×106
Optimiser Batch size, weight decay Schedule EMA decay, gradient clip Early-stopping patience
AdamW, lr 3×10−4 512, 10−5 cosine, 5-epoch warmup 0.999, 1.0 15 epochs
4.3 Evaluation We evaluate the trained cINN in two complementary regimes. In open-loop evaluation, the network sees groundtruth state–command pairs (st , ct ) from the held-out test set and we compare its predicted motor command against the ground-truth INDI command at the same time-step; the simulator is not in the loop. In closed-loop evaluation, the network replaces INDI in the inner control loop: at every 10 ms step the simulator state is read out, ct is built from the current velocity command and tracking errors, and the predicted motor command is written back to the plant. The two regimes thus probe different properties: open-loop quantifies one-step prediction accuracy and calibration; closed-loop reveals whether the small per-step errors compound into stable or unstable trajectories. Open-loop evaluation on the 125 100 held-out test samples reports the coefficient of determination R2 , root-meansquare error (RMSE), and mean absolute error (MAE) of the mode prediction ût = fθ−1 (0; h), both per channel and aggregated. Probabilistic quality is assessed via the continuous ranked probability score (CRPS) computed from N = 20 samples per test point and the empirical coverage at 68 % and 95 % nominal levels; we apply a single post-hoc temperature T minimising the validation NLL. Closed-loop evaluation uses 15 held-out scenarios disjoint from training. For each, we run the simulator with INDI and with the cINN replacing INDI in the inner loop, and log position RMSE, final position error, peak pitch excursion, and episode duration up to either nominal termination or the simulator’s attitude safety break. A single inference takes 47.5 ms on the RTX 3090, longer than the 10 ms control cycle, so the evaluation uses the simulator’s time abstraction to step the cINN deterministically once per control step.
s_t
12-D
sin/cos
s̃ _t
η: 3 → 6 pos, vel, ω (9-D)
concat
15-D
24-D
Encoder MLP 3 × 256, SiLU
h
256-D
c_t
9-D
conditions every coupling
K = 9 stacked invertible blocks u
ActNorm
Spline coupling
Inv linear
z
8-D
data-init
split 4+4, 8 bins
8×8 mixing
8-D
training: u → z, log p(u | s_t, c_t) → NLL
inference: z ∼ N(0, I), sample u
Fig. 1. cINN architecture. The encoder maps the augmented state s̃t and command ct to a context h, which conditions each of K = 9 flow blocks. Each block consists of activation normalisation, rational-quadratic spline coupling, and a learned invertible linear map; the base distribution is N (0, I8 ). 5. RESULTS 5.1 Open-Loop Reproduction Across 125 100 held-out test samples the cINN reproduces the INDI teacher with mean R2 = 0.944, RMSE 121.0 rad/s (7.8 % of the operating range), and MAE 62.0 rad/s. Per-channel R2 stays within 0.940–0.950; the small asymmetry between Motors 1, 7 and Motors 2, 8 reflects the diagonal pairing of coaxial rotors in the XZplane. Aggregate metrics are in Table 2; Fig. 2 shows the prediction vs. ground-truth relationship for the two motors that bracket the per-channel range. Table 2. cINN open-loop performance over 8 motor channels and 125 100 test samples (mean and range across motors). Inference: 47.45 ms per sample on RTX 3090. Metric
Mean
Range
RMSE [rad/s] RMSE [% range] MAE [rad/s] R2 CRPS Coverage 68 % Coverage 95 % NLL
121.0 7.8 62.0 0.944 0.0915 57.7 % 73.9 % 34.76
116.8–127.6 7.2–8.3 58.2–64.9 0.940–0.950 0.0858–0.0974 56.0–59.0 % 72.1–75.6 % ±15.03
Fig. 3 shows the calibration curve, comparing the nominal coverage (the probability mass the model places inside a central predictive interval) against the empirical coverage (the fraction of test samples that fall inside it). A wellcalibrated model lies on the diagonal. The cINN stays systematically below it on the upper half, reaching 57.7 % at 68 % nominal and 73.9 % at 95 % nominal—mild overconfidence rather than underconfidence; mean CRPS is 0.0915 (post-hoc temperature T = 1.214). Log-probability and
Fig. 2. Predicted vs. ground-truth motor commands for Motor 1 (R2 = 0.940, per-channel worst) and Motor 2 (R2 = 0.950, per-channel best); other motors fall between these two. absolute prediction error correlate at Pearson ρ = −0.60: low-confidence samples are those the model tends to get wrong. The flow stays numerically invertible (round-trip reconstruction mean error 2.8×10−5 in standardised units). 5.2 Closed-Loop Deployment Across 15 held-out scenarios the cINN reaches mean position RMSE 9.7 m (INDI: 9.5 m) and final position error 8.8 m (INDI: 8.7 m). The aggregate peak pitch of 42◦ (cINN) vs. 13◦ (INDI) conflates two regimes: in successful scenarios the cINN stays inside the INDI envelope, while failure scenarios trigger the simulator’s attitudesafety break and dominate the maximum. The per-episode distribution (Fig. 4) is bimodal. We sort the scenarios into four tiers: success ({4, 5, 8}), acceptable ({7, 10, 12, 15}),
Fig. 3. Calibration curve of the open-loop predictions. The model is mildly overconfident at high nominal levels.
Fig. 5. Closed-loop response for Scenario 1 (Tier D, phaselag failure).
(a) trajectory Fig. 4. Closed-loop comparison over 15 scenarios. (a) Position RMSE per episode. (b) Aggregate metrics. poor ({3, 6, 9, 13}), and failure ({1, 2, 11, 14}); 7/15 ≈ 47 % are success or acceptable. 5.3 Success Cases In the successful and acceptable scenarios, the cINN remains within the same qualitative operating envelope as the INDI baseline. These cases are characterised by slowly varying velocity references and pitch excursions that stay below approximately ±20◦ . In this regime, which is well covered by the training distribution, the cINN follows the INDI trajectory with comparable settling behaviour and without triggering the simulator’s attitude-safety break. The success cases also reveal a consistent difference between the learned controller and the teacher. While the mean motor command remains close to the INDI command level, the cINN output contains more high-frequency content. This additional variation does not destabilise the plant in the successful scenarios, but it indicates that the learned inverse-dynamics model does not reproduce the filtering and smoothness properties of the teacher perfectly. 5.4 Failure Modes The four failure scenarios separate into two distinct mechanisms. Command-tracking phase lag (Scenario 1, Fig. 5): the velocity command alternates between +0.5 and −0.5 m/s
(b) pitch
Fig. 6. Attitude divergence in Scenario 11. The cINN reaches 90◦ pitch within 1 s; INDI completes the same reference stably. at roughly 1.5 s intervals. The cINN remains stable over the full 12.5 s episode—peak pitch below 3◦ and mean motor command at the INDI level—but the velocity response (c, d) trails INDI by about one switching period, and position error (b) accumulates as an oscillation rather than diverging. The lag follows from the conditioning structure: ct encodes the instantaneous error but not its rate of change, so the model cannot anticipate a direction reversal. Attitude divergence (Scenarios 2, 11, 14, Fig. 6): within the first two seconds an aggressive velocity step starting from non-zero pitch drives the cINN to a large motor differential, the pitch crosses ±30◦ , and the simulator’s attitude safety break terminates the episode. INDI handles the same references stably, so the failure is specific to the learned controller. The common feature is a rapid command change from a non-zero attitude—a combination underrepresented in the training distribution, and the regime in which the predictive log-probability is lowest, consistent with the ρ = −0.60 correlation above. 6. CONCLUSION We trained a conditional invertible neural network as a probabilistic inverse-dynamics model for a 2-D X8 coaxial multicopter and evaluated it against an INDI teacher in open- and closed-loop. Open-loop, the cINN reproduces the teacher’s eight-channel motor commands at R2 =
0.944. Closed-loop, it reaches a comparable mean position RMSE to INDI over 15 held-out scenarios, but with bimodal behaviour: 47 % of scenarios are tracked successfully or acceptably, while the remaining cases expose clear failure modes. The probabilistic output is informative but not fully calibrated: empirical coverage remains below the nominal levels at high coverage, indicating mildly overconfident predictive intervals. Nevertheless, log-probability correlates with absolute prediction error at ρ = −0.60, giving a useful runtime confidence signal. The two observed failure modes suggest direct extensions: temporal context or command derivatives to reduce phase lag, and targeted data generation for aggressive commands from non-zero attitude. A hybrid controller, in which the cINN acts only above a confidence threshold and INDI takes over otherwise, is the most immediate route towards safer deployment. Full 6-DoF flight and hardware deployment further require onboard state estimation and reducing the current 47.5 ms inference time. DECLARATION OF GENERATIVE AI The authors used DeepL Write for language editing and Claude (Anthropic) for code-related assistance. All content was reviewed and edited by the authors, who take full responsibility for the publication. REFERENCES Ardizzone, L., Kruse, J., Wirkert, S., Rahner, D., Pellegrini, E.W., Klessen, R.S., Maier-Hein, L., Rother, C., and Köthe, U. (2019). Analyzing inverse problems with invertible neural networks. In International Conference on Learning Representations. Chao, H., Cao, Y., and Chen, Y. (2010). Autopilots for small unmanned aerial vehicles: A survey. International Journal of Control, Automation and Systems, 8(1), 36– 44. doi:10.1007/s12555-010-0105-z. Cheng, G. et al. (2024). Physics-informed neural network for quadrotor dynamical modeling. Robotics and Autonomous Systems, 171, 104569. doi: 10.1016/j.robot.2023.104569. Denker, A., Schmidt, M., Leuschner, J., and Maass, P. (2021). Conditional invertible neural networks for medical imaging. Journal of Imaging, 7(11), 243. doi: 10.3390/jimaging7110243. Dierks, T. and Jagannathan, S. (2009). Neural network control of quadrotor UAV formations. In Proceedings of the American Control Conference, 2990–2996. doi: 10.1109/ACC.2009.5160591. Dinh, L., Sohl-Dickstein, J., and Bengio, S. (2017). Density estimation using Real-NVP. In International Conference on Learning Representations. Durkan, C., Bekasov, A., Murray, I., and Papamakarios, G. (2019). Neural spline flows. In Advances in Neural Information Processing Systems, volume 32, 7509–7520. Foehn, P., Romero, A., and Scaramuzza, D. (2021). Time-optimal planning for quadrotor waypoint flight. Science Robotics, 6(56), eabh1221. doi: 10.1126/scirobotics.abh1221. Heredge, J., Sheridan, S., et al. (2024). Generative invertible quantum neural networks. SciPost Physics, 16, 146. doi:10.21468/SciPostPhys.16.6.146.
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