Quantasio: A Generalized Neural Framework for Solving 3D Navier-Stokes Dynamics

Welcome to Quantasio! This tool leverages deep learning to approximate solutions for any general temporal PDE as well as fluid dynamics problems using the Navier-Stokes equations. The solver is designed to handle 2D, 3D, and 4D (time-dependent) cases and is capable of solving a range of scenarios from basic flows to turbulent regimes and flows around obstacles.

Features

General PDE Solver:

Supports solving temporal problems in:

1. 2D: Solves equations of the form u(x, t).
2. 3D: Extends to u(x, y, t).
3. 4D: Extends to u(x, y, z, t).
4. NS Solver: Approximates vector fields (u, v, w)(x, y, z, t) by solving the Navier Stokes Equation.

Navier-Stokes Applications:

Solves the Navier-Stokes equations for incompressible fluids. Handles boundary conditions such as inflow, outflow, and no-slip surfaces.

Scenarios:

Basic Navier-Stokes Problem: Steady or unsteady flows in a regular domain. Turbulent Flow: Approximates turbulent-like behavior using adjusted initial and boundary conditions. Flow Around Obstacles: Simulates flows in domains with internal obstacles, such as a sphere. Visualization:

Visualize the solutions with animations showing the evolution of velocity fields over time. Support for rendering vector fields in 2D, 3D, and 4D domains with obstacles.

Case Study 1: Visualizing a Basic Flow field by solving NS Equation

For a velocity field u = (u, v, w) in a 3D domain:

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} - \nu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right) = 0$$ $$\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} - \nu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right) = 0$$ $$\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} - \nu \left(\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right) = 0$$

Where:

• $( u, v, w )$: Velocity components in $(x)$, $(y)$, and $(z)$ directions.
• $( \nu )$: Kinematic viscosity.

Initial Conditions:

The initial velocity field is specified as:

$$u_0(x, y, z) = \sin(\pi x), \quad v_0(x, y, z) = 0, \quad w_0(x, y, z) = 0$$

Domain:

The domain is defined as:

$$x \in [0, 1], \quad y \in [0, 1], \quad z \in [0, 1], \quad t \in [0, 1]$$

Residuals:

The deep learning model minimizes the residuals of the Navier-Stokes equations, which are defined as:

$$f_u = \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} - \nu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right)$$ $$f_v = \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} - \nu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right)$$ $$f_w = \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} - \nu \left(\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right)$$

Visualizing and animating the flow field using Quantasio

Case Study 2: Visualizing a Turbulent flow field with oscillatory boundary conditions

Initial Conditions:

The initial velocity field is given by sinusoidal variations:

$$u_0(x, y, z) = \sin(\pi x) \cos(\pi y)$$ $$v_0(x, y, z) = \sin(\pi y) \cos(\pi z)$$ $$w_0(x, y, z) = \sin(\pi z) \cos(\pi x)$$

Oscillatory Boundary Conditions:

At the domain boundaries, the velocity field exhibits oscillatory behavior:

$$u_b(x, y, z, t) = 0.5 \sin(2\pi t) \cos(\pi x)$$ $$v_b(x, y, z, t) = 0.5 \sin(2\pi t) \cos(\pi y)$$ $$w_b(x, y, z, t) = 0.5 \sin(2\pi t) \cos(\pi z)$$

Domain:

The domain is defined as:

$$x \in [0, 1], \quad y \in [0, 1], \quad z \in [0, 1], \quad t \in [0, 1]$$

Visualizing and animating the flow field using Quantasio

Case Study 3: Visualizing Turbulent flow over a spherical obstacle

Initial Conditions:

The velocity field is initially zero throughout the domain:

$$u_0(x, y, z) = 0, \quad v_0(x, y, z) = 0, \quad w_0(x, y, z) = 0$$

Boundary Conditions:

1. Inflow (Uniform Flow):

$$u(x, 0, z, t) = 1, \quad v(x, 0, z, t) = 0, \quad w(x, 0, z, t) = 0$$

2. No-slip condition on obstacle surface: On the spherical obstacle centered at $( (0.5, 0.5, 0.5) )$ with radius $( r = 0.1 )$,

$$u = v = w = 0$$

3. Outflow (Zero-gradient boundary condition): At the outflow boundary, the velocity gradients are zero.

Domain:

The computational domain is defined as:

$$x \in [0, 1], \quad y \in [0, 1], \quad z \in [0, 1], \quad t \in [0, 1]$$

Obstacle Representation:

The obstacle is represented as a sphere:

$$(x - 0.5)^2 + (y - 0.5)^2 + (z - 0.5)^2 \leq 0.1^2$$

Visualizing and animating the flow field using Quantasio

📄 Citation

Paper Link

If you find this repository useful in your research, please cite my work as follows:

@article{sarker2025quantasio,
  title={Quantasio: A Generalized Neural Framework for Solving 3D Navier-Stokes Dynamics},
  author={Sarker, Soumick and Chakraborty, Sudipto},
  journal={IEEE Transactions on Artificial Intelligence},
  year={2025},
  publisher={IEEE}
}