No, in this paper we enforced the Lagrangian structure is enforced as a hard constraint, via the model architecture. It's not just a loss term. Soft constraint

No, in this paper we enforced the Lagrangian structure is enforced as a hard constraint, via the model architecture. It's not just a loss term. Soft constraints via loss functions can also help, but in my experience they are much less effective than hard constraints. My impression is that this is pretty broadly consistent with the experience of others working in this field. For neural nets with 3D invariance, I would strongly recommend looking into the literature on "group equivariant convolutions". This has been very active area of research over the past few years, e.g., see the work of Taco Cohen: https://arxiv.org/abs/1902.04615 https://arxiv.org/abs/1902.04615