  We delve into the physics-informed neural reconstruction of smoke and
obstacles through sparse-view RGB videos, tackling challenges arising from
limited observation of complex dynamics. Existing physics-informed neural
networks often emphasize short-term physics constraints, leaving the proper
preservation of long-term conservation less explored. We introduce Neural
Characteristic Trajectory Fields, a novel representation utilizing Eulerian
neural fields to implicitly model Lagrangian fluid trajectories. This
topology-free, auto-differentiable representation facilitates efficient flow
map calculations between arbitrary frames as well as efficient velocity
extraction via auto-differentiation. Consequently, it enables end-to-end
supervision covering long-term conservation and short-term physics priors.
Building on the representation, we propose physics-informed trajectory learning
and integration into NeRF-based scene reconstruction. We enable advanced
obstacle handling through self-supervised scene decomposition and seamless
integrated boundary constraints. Our results showcase the ability to overcome
challenges like occlusion uncertainty, density-color ambiguity, and
static-dynamic entanglements. Code and sample tests are at
\url{https://github.com/19reborn/PICT_smoke}.
