In a computer simulation of canal flow, a reinforcement-learning controller seeded with prior knowledge from analytical backstepping remained stable when the starting water-depth and discharge profiles were far from their target values. In the reported large-deviation case, TDPSAC was the best-performing of the learning-based controllers. Analytical backstepping failed to maintain convergence in that test, while all the learning-based methods remained stable. The finding is limited to the tested model and scenario: it does not establish that the controller will behave the same way in a physical canal.
The work is an arXiv preprint, version 1 dated 20 Aug 2026. It asks whether backstepping control knowledge embedded in a SAC system can improve learning efficiency and nonlinear canal-flow regulation. The evaluation used numerical finite-difference simulations of a Saint-Venant canal with Sambre River parameters, rather than observations from people, animals or a field trial.
Borrowing an analytical control map
TDPSAC, short for Transferred DeepONet Prior SAC, uses pretrained DeepONet modules to build SAC representations. Selected layers are frozen and fine-tuning rates reduced to preserve prior knowledge, while the applied input combines the SAC controller with the learned prior. The design therefore carries information from analytical control into the learning system as it adapts.
The prior was learned in a separate pretraining step. DeepONet was trained to approximate the analytical backstepping state-feedback map—the rule that links the current canal state to a control action—using closed-loop Saint-Venant trajectories. The trajectories began from conditions perturbed around equilibrium with Chebyshev functions. In ordinary terms, the learning system was first shown examples of how the analytical controller responds before the SAC component was fine-tuned.
The pretraining design set the truncated Chebyshev expansion parameter M to 6. It allowed maximum perturbations of 0.30 metres in water depth and 2.50 cubic metres per second in discharge. Only trajectories that converged under analytical backstepping were retained, leaving 105 training samples. That filter meant the prior was built from converging examples rather than every possible starting condition.
The pretrained DeepONet's average L2 approximation error—an overall measure of the difference between its output and the analytical control map—was less than 10^-4 over 50 evaluations. In the ablation comparison, TDPSAC also achieved the highest and most stable return among learning-based controllers. Return is the reward score used to track performance during reinforcement-learning training.
The difference appeared when the starting point moved
When the simulated canal started near equilibrium, all controllers regulated the system. Analytical backstepping produced the smallest steady-state errors, but its transient overshoots were larger. The learning-based controllers had smaller overshoots. The result presents a split performance picture: backstepping was strongest on final errors in this case, while the learning-based methods were gentler during the adjustment.
To test both cases, the study used 50 randomly sampled trials for each initial-deviation setting. For both water depth H and discharge Q, 25 samples began above equilibrium and 25 below it. The reported table summarizes results as means plus or minus standard deviations across those 50 trials; no p-values or formal inferential tests were reported.
In Case I, the near-equilibrium row for TDPSAC reported a water-depth profile error ϵH of 2.72 ± 0.0261 and a discharge-profile error ϵQ of 8.78 ± 0.614. Its maximum positive relative overshoot measures were 0.947 ± 0.969 for water depth and 28.0 ± 11.6 for discharge. The table reported tRL as 41. These are descriptive values across the trial set, not a claim that the same figures will hold outside the simulation.
The larger-deviation case began with water-depth deviation ϵH of 6.3% and discharge deviation ϵQ of 20%. In Case II, TDPSAC's reported water-depth and discharge error values were 4.57 ± 0.276 and 16.9 ± 4.35. Its maximum positive relative overshoots were 3.77 ± 3.81 for water depth and 98.4 ± 56.3 for discharge, and tRL was 41. BS was absent from the table because it failed to converge.
A result with a narrow boundary
Across the learning-based controllers, reported RL training times ranged from 32 to 41 minutes. TDPSAC's tRL value was 41 in both table cases. The times describe the training runs in this study; they do not by themselves establish an operational advantage.
The numerical setting used a spatial step of 111.28 metres, a time step of 5 seconds and a simulation horizon of 666.67 minutes. Every comparison—including the near-equilibrium and large-deviation results—therefore sits inside that finite-difference model and its chosen time and space resolution.
The boundary of the evidence is important. The study uses one Sambre River parameterization and numerical trajectories, with no physical-canal or field validation reported in the supplied analysis. It also reports no formal global stability proof or guarantee for TDPSAC. Persistent disturbances and slowly varying operating conditions were not evaluated. Those omissions leave open whether the result would survive conditions that were not represented in the tested simulation.
More testing would be needed to learn whether the reported advantage generalizes to other canal parameterizations and untested initial profiles, and whether the learned controller can be given formal stability or robustness guarantees. For now, the preprint's strongest claim is narrower: within the reported numerical setup, transferred backstepping knowledge was associated with more efficient learning and better large-deviation regulation than the tested learning-based alternatives.
The paper reports support from the National Natural Science Foundation of China under grant 62173084.
Paper data and sources
Original title: Backstepping-Guided Reinforcement Learning for Wide-Range Saint-Venant Canal Regulation
Authors: Chenchen Wang, Jie Qi
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text