# Physics-Informed Learning of Characteristic Trajectories for Smoke Reconstruction

arXiv: https://arxiv.org/abs/2407.09679 · HTML: https://mlsanity.com/paper/2407.09679/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 63 references that are in the map.
- First posted on arXiv 12 Jul 2024.

## Builds on (most cited first, 29 of 29)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields](https://mlsanity.com/paper/2103.13415/) (2021)
- [NeuS: Learning Neural Implicit Surfaces by Volume Rendering for Multi-view Reconstruction](https://mlsanity.com/paper/2106.10689/) (2021)
- [NeRF in the Wild: Neural Radiance Fields for Unconstrained Photo Collections](https://mlsanity.com/paper/2008.02268/) (2020)
- [PIFu: Pixel-Aligned Implicit Function for High-Resolution Clothed Human Digitization](https://mlsanity.com/paper/1905.05172/) (2019)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [Neural Scene Flow Fields for Space-Time View Synthesis of Dynamic Scenes](https://mlsanity.com/paper/2011.13084/) (2020)
- [Non-Rigid Neural Radiance Fields: Reconstruction and Novel View Synthesis of a Dynamic Scene From Monocular Video](https://mlsanity.com/paper/2012.12247/) (2020)
- [Dynamic Neural Radiance Fields for Monocular 4D Facial Avatar Reconstruction](https://mlsanity.com/paper/2012.03065/) (2020)
- [Dynamic View Synthesis from Dynamic Monocular Video](https://mlsanity.com/paper/2105.06468/) (2021)
- [NeuS2: Fast Learning of Neural Implicit Surfaces for Multi-view Reconstruction](https://mlsanity.com/paper/2212.05231/) (2022)
- [Neural Radiance Flow for 4D View Synthesis and Video Processing](https://mlsanity.com/paper/2012.09790/) (2020)
- [HyperNeRF: A Higher-Dimensional Representation for Topologically Varying Neural Radiance Fields](https://mlsanity.com/paper/2106.13228/) (2021)
- [AutoInt: Automatic Integration for Fast Neural Volume Rendering](https://mlsanity.com/paper/2012.01714/) (2020)
- [Physics Informed Neural Fields for Smoke Reconstruction with Sparse Data](https://mlsanity.com/paper/2206.06577/) (2022)
- [Implicit Neural Spatial Representations for Time-dependent PDEs](https://mlsanity.com/paper/2210.00124/) (2022)
- [ResFields: Residual Neural Fields for Spatiotemporal Signals](https://mlsanity.com/paper/2309.03160/) (2023)
- [Global Transport for Fluid Reconstruction with Learned Self-Supervision](https://mlsanity.com/paper/2104.06031/) (2021)
- [Inferring Hybrid Neural Fluid Fields from Videos](https://mlsanity.com/paper/2312.06561/) (2023)
- [Fluid Simulation on Neural Flow Maps](https://mlsanity.com/paper/2312.14635/) (2023)
- [ScalarFlow: A Large-Scale Volumetric Data Set of Real-world Scalar Transport Flows for Computer Animation and Machine Learning](https://mlsanity.com/paper/2011.10284/) (2020)
- [GDOD: Effective Gradient Descent using Orthogonal Decomposition for Multi-Task Learning](https://mlsanity.com/paper/2301.13465/) (2023)

## Built on by (most cited first, 9 of 9)

- [PhysCtrl: Generative Physics for Controllable and Physics-Grounded Video Generation](https://mlsanity.com/paper/2509.20358/) (2025)
- [FluidNexus: 3D Fluid Reconstruction and Prediction from a Single Video](https://mlsanity.com/paper/2503.04720/) (2025)
- [FreeGave: 3D Physics Learning from Dynamic Videos by Gaussian Velocity](https://mlsanity.com/paper/2506.07865/) (2025)
- [Physics-Informed Deformable Gaussian Splatting: Towards Unified Constitutive Laws for Time-Evolving Material Field](https://mlsanity.com/paper/2511.06299/) (2025)
- [Representing Flow Fields with Divergence-Free Kernels for Reconstruction](https://mlsanity.com/paper/2504.01913/) (2025)
- [LagrangianSplats: Divergence-Free Transport of Gaussian Primitives for Fluid Reconstruction](https://mlsanity.com/paper/2605.09299/) (2026)
- [SmokeSVD: Smoke Reconstruction from A Single View via Progressive Novel View Synthesis and Refinement with Diffusion Models](https://mlsanity.com/paper/2507.12156/) (2025)
- [DyG2T: Modeling Object Dynamics with 3D Gaussian Temporal-Spatial Particle Graph Transformer](https://mlsanity.com/paper/2608.18498/) (2026)
- [SplashSplat: Reconstructing Splashing Liquids from Real-World Multi-View Videos](https://mlsanity.com/paper/2609.20818/) (2026)

## Abstract (arXiv)

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 urlhttps://github.com/19reborn/PICTsmoke.