1. Although the use of this type of loss in this setting might be new, this work does not prove any new theoretical results.
2. That being said, experiment is a very important component in this paper, however, I find the evaluation metric of the solution very interesting. More specifically, let $u$ be the output of neural networks and $u^*$ be the exact solution. The test error is usually computed using relative $L^2$ norm (See for example [1][2]), i.e.
$$|| u - u^*||_2^2 / ||u^*||_2^2 = \int|u - u^*|^2dx / \int |u^*|^2 dx.$$
However, in Figure 4, when evaluating solutions, the mean error is computed using equation (15), the energy norm.
(i). why not using the relative $L^2$ norm? How does Astral loss perform if the evaluation is done in $L^2$?
(ii). The a posteriori error bound is in the energy norm, i.e.
$$L(u, w_L) \leq |||u-u^*||| \leq U(u, w_U).$$
so I would naturally expect Astral loss to achieve fairly small error in this energy norm, but this does not necessarily imply the solution is "better". Equations can be solved in different spaces. In fact, I think the space $L^2$ is more commonly used when people study existence and uniqueness of PDE solutions.
(iii). There could be a relation between the energy norm and $L^2$ norm. More explanation is needed for the specific choice of the evaluation metric since it differs from the previous literature.
[1] Li et al., Physics-Informed Neural Operator for Learning Partial Differential Equations
[2] Wang et al., An Expert's Guide to Training Physics-informed Neural Networks
1. Although the use of this type of loss in this setting might be new, this work does not prove any new theoretical results.