### Foundations of Interpretable and Reliable Machine Learning

Question we address: How to develop physics-informed reinforcement learning algorithms that guarantee safety and interpretability ?

It is widely known that policies trained using reinforcement learning (RL) to solve simulated robotics problems (MuJoCo) are extremely brittle and unstable, i.e. your solution will most likely break down after perturbing a bit (e.g. poking the robot) or transferring it to a similar task. It is often impossible to provide any safety guarantees for constraint satisfaction or an interpretation of how the trained policies work.

##### Challenges in Theory
There are many opportunities for work on theoretical foundations of the developing SPP approach. This area is yet unexplored, we present below the most promising research directions aiming at important theorems.
Work towards theorems providing convergence guarantees of the Bellman iterates for our variant of the Q-function that involves an inverse dynamics control model.
Establish link between State Planning Policies and Hamilton-Jacobi-Bellman optimization in order to compute safety guaranteeing policies;
Formal Verification -- we can attempt a formal verification of trained policies, i.e. extract from the neural network rules and mathematically verify that the policy will satisfy the safety constrains within some fixed time horizon;

$Q^{\pi, CM}(s_t, a_t) = \mathbb{E}_{r_{i\ge t},s_{i>t}\sim E,\ z_{i>t}\sim \pi,\ a_i = CM(s_i,z_i)}{\left[R_t|s_t,a_t\right]};$
$0 = \min\left\{l(x)-V(x,t), \frac{\partial V}{\partial t}+\max_{u\in\mathcal{U}}{\nabla_x{V^Tf(x,u)}}\right\}.$