Guaranteed Neural PDE Boundary Control with Neural Barrier Function

1. The intuition for including the filtering threshold $\eta$ is not clear to me, as it would seem to allow unsafe behavior when $\dot{U}_{nominal}$ is far from being safe. Could you please explain why you include this threshold?
2. You say that you collect training data using the pre-trained RL controllers. Is there any risk that replacing the nominal controller with the filtered controller will cause a distribution shift that creates an error in your learned transfer function & BCBF? How would you mitigate this error? Could you instead use some other control input (e.g. persistently exciting) to gather training data?
3. In Tables 1 and 2, why did you use different boundary feasibility constraints for PPO and SAC?
4. Could you comment on why you think the feasibility rate is not 100%? Is the BCBF not fully trained? Is there model mismatch?
5. "because time-independent BCBF tends to have divergent performance with more non-feasible trajectories and more feasible steps for feasible trajectories" -> why do you think this is?
6. "showing that the strong safety filtering may hurt the stability of the PPO controller due to the model mismatch between direct boundary mapping with the neural operator and underlying PDE dynamics" -> Is this because the neural operator is trained only using the nominal controller? Could you learn these dynamics using a random control input and get better results with less mismatch? How do you know that filtering hurts the stability of the controller; could it instead be that high reward is not compatible with feasibility on this problem?

In equation (7), "$]_+$" is missed.

The QP in (13) seems to be a single-variable convex problem that can be solved analytically. It is unclear why it cannot be solved in real-time.

Learning the model $Y(t)=G(U(t))$ is generally hard, and sample-complex for PDEs. It will be interesting to see how the learned models behaviors for high-dimensional PDE beyond 1-D and 2-D.

In Section 4.1, how is the data $(U(t), Y(t))$ collected? by solving the PDE? That might be computationally expensive and inapplicable for high-dimensional PDEs

The experimental results are confusing to interpret/explain, listed as follows.

In Table 1, the PPO baseline without a safety filter does not achieve the highest reward, which does not read right. Also, the rewards in this table are super noisy with large std. Considering that all three approaches (either the PPO ones or SAC ones) do not show a clear margin between them. That is probably why PPO/ SAC without safety filters do not achieve the highest reward. So I am wondering why the reward is so noisy.

Also in Table 1, for PPO approaches, why did the time-independent filter $\phi(Y)$ failed to improve the feasible rate? For SAC, the time-independent filter $\phi(Y)$ even lowers the feasible rate significantly, casting doubts on the effectiveness of this approach, or at least, the implementation.

Problems similar to those in Table 1 exist in Tables 2 and 3.

In Table 3, the tradeoff between reward and feasibility is not significant. It reads like without any loss of high reward, the feasible rate is improved by 13 percent.

It is not clear why "Partial derivative equations (PDEs)" is used while "partial differential equation" sounds more common.

N/A.