# Physics Informed Neural Fluid Fields

## Supplemental Webpage

#### We provide results in form of this html page, so that individual scenes can be easily and repeatedly watched. Please click on each scene to play the corresponding video. It is helpful to change the zoom level of the browser (Ctrl and +/-) or to view in full screen (double click on videos or right click and open in new view). All videos below are located in the video sub-folder.

This document contains the following parts:
 1. ScalarFlow Datasets    1.1 Synthetic Scene    1.2 Real Captures 2. Regular Fluids with Complex Lighting    2.1 Regular Plume Scene    2.2 Scenes with Regular Obstacles 3. Complex Scenes with Arbitrary Obstacles 4. Training Details

## 1. ScalarFlow Datasets

#### -- 1.1 ScalarFlow Synthetic Smoke --

Rendering Comparisons

#### Taking multiview videos as input, our method learns the spatio-temporal density, color, and velocity fields. "Rendering Comparisons" (left) shows the overall quality of density and color together. While the reference has a black background, we render reconstructions with a blue background, so that "ghost" density in the color of the original background is visible. "Volumetric Density" (below) shows the density alone using a uniform ambient light. "Velocity" and "Vorticity" (further below) are learned implicitly though density fields. Note that only GlobalTrans requires lighting conditions and visual hulls as input. Other methods with unknown lighting conditions have to disentangle the ambiguity between density and color. - Our rendering result is close to the reference. - Neural Volumes has artifacts on novel views due to the existence of "ghost density". - Although Global Transport (GlobalTrans) produces sharp results, it does not faithfully reconstruct the actual scene, but instead has a lot of noise in both density and velocity.

 Volumetric Density, (front-side-top, rendered with uniform ambient lighting) Warping Errors, (left: warp frame i to i+1; right: warp frame i and i+1 to i+0.5)

#### - GlobalTrans-Warp-Error: GlobalTrans has minimal full-step warping error. It is trained at this discrete level of time. - Ref-Warp-Error: Simulated with a time step of 0.5, the reference has a high numerical error when warping with a time step of 1.0 - Ours-Warp-Error and Ours-MidWarp-Error: Our continuous model has slightly larger full-step warping error and minimal mid-step warping error. Note that we use the same training data as GlobalTrans without mid-step frames being observed.

 Velocity, (the middle slice of front-side-top, intensity reduced outside visual hull) Vorticity, (the middle slice of front-side-top within the visual hull)

#### -- 1.2 Real Smoke Captures --

 Rendering Comparisons Volumetric Density, (front-side-top) Velocity(left) and Vorticity(right), (middle slices, front-side-top) Warping Error, (left: warp frame i to i+1; right: warp frame i and i+1 to i+0.5)

## 2. Synthetic Scenes with Complex Lighting

#### -- 2.1 The Plume Scene with Velocity Comparisons --

Rendering Comparisons

#### - NeuralVolumes: With much "ghost density", NeuralVolumes can easily render more details, since keeping view-consistency is not necessary with their occlusion. - Deformation: Representing turbulent fluid dynamics as deformation fields is ill-posed and results in stretches due to rigid motion. - Ours w.o. d2v: Comparing to NeuralVolumes, Ours w.o. d2v significantly reduces the "ghost density", but has color-bleeding artifacts due to less accurate velocity fields - Ours: Our results match the reference in both training and novel views best with properly disentangled density and color.

Volumetric Comparisons

#### Comparisons with Related Work

Sphere Scene, Ref Ours Ours w.o. d2v NeRF+T Neural Volumes

#### Our Results

Unsupervised Separation of Static and Dynamic parts

#### Static and dynamic components are nicely seperated.

Estimated Density and Velocity

Car Game

## 4. Training Details

#### Across all scenes, we use the Adam optimizer with a learning rate of 0.0001. Other details of each scene are given in the following table (using a single NVIDIA Quadro RTX 8000 GPU):

Scenes Image Resolution Total Training
Iterations
Total Training
Time
Hyper-parameters for
Hyper-parameters for
Velocity Supervision
ScalarFlow Synthetic 360x640 200k 30h $\mathcal{L}_{\widetilde{\mathit{img}}} + 0.025\mathcal{L}_{VGG} + 0.1\mathcal{L}_{ghost}$ $2\mathcal{L}_{\frac{D\sigma}{Dt}} + 0.0005\mathcal{L}_{NSE} + 6\mathcal{L}_{d2v}$
Real 540x960 500k 74h $\mathcal{L}_{\widetilde{\mathit{img}}} + 0.025\mathcal{L}_{VGG} + 0.1\mathcal{L}_{ghost}$ $2\mathcal{L}_{\frac{D\sigma}{Dt}} + 0.0005\mathcal{L}_{NSE} + 6\mathcal{L}_{d2v}$
Complex
Lighting
Plume 400x400 200k 31h $\mathcal{L}_{\widetilde{\mathit{img}}} + 0.025\mathcal{L}_{VGG} + 0.05\mathcal{L}_{ghost}$ $2\mathcal{L}_{\frac{D\sigma}{Dt}} + 0.0005\mathcal{L}_{NSE} + 6\mathcal{L}_{d2v}$
Sphere 400x400 150k 37h $\mathcal{L}_{\widetilde{\mathit{img}}} + 0.025\mathcal{L}_{VGG} + 0.05\mathcal{L}_{ghost} + 0.05\mathcal{L}_{overlay}$ $2\mathcal{L}_{\frac{D\sigma}{Dt}} + 0.0005\mathcal{L}_{NSE} + 6\mathcal{L}_{d2v}$
Complex
Obstacles
Car 960x500 200k 51h $\mathcal{L}_{\widetilde{\mathit{img}}} + 0.025\mathcal{L}_{VGG} + 0.01\mathcal{L}_{ghost} + 0.05\mathcal{L}_{overlay}$ $2\mathcal{L}_{\frac{D\sigma}{Dt}} + 0.0005\mathcal{L}_{NSE} + 6\mathcal{L}_{d2v}$
Game 800x800 250k 64h $\mathcal{L}_{\widetilde{\mathit{img}}} + 0.025\mathcal{L}_{VGG} + 0.01\mathcal{L}_{ghost} + 0.05\mathcal{L}_{overlay}$ $2\mathcal{L}_{\frac{D\sigma}{Dt}} + 0.0005\mathcal{L}_{NSE} + 6\mathcal{L}_{d2v}$