How to Use a PINN for a Navier-Stokes Inverse Problem

A from-scratch PyTorch build recovers blood flow, viscosity, and wall shear stress in a narrowed artery from 40 noisy velocity readings.

Wall shear stress is the friction blood puts on the wall of a vessel. It is tied to where plaque builds up, and it is hard to measure directly. A physics-informed neural network (PINN) is a natural fit. I built one in plain PyTorch, without DeepXDE or any other PINN library, for a 2D artery with a narrowing (a stenosis). From 40 velocity readings it reconstructs the velocity and pressure fields and the region of reversed flow behind the narrowing. It also works out the viscosity of the fluid, which I treated as unknown, and its wall shear stress follows the CFD reference closely.

The artery is a 2D channel of height H with a smooth bump on the lower wall that blocks half of the opening, a 50% stenosis. The flow is incompressible Navier-Stokes at a Reynolds number of 200. In these units the kinematic viscosity is 0.005. To score a reconstruction you need to know the right answer, so I generated it. A Navier-Stokes solver I wrote computes the steady flow, and the PINN only ever gets a few noisy readings taken from that solution. The noise is Gaussian, at 7% of the inlet velocity.

The network is a plain MLP with 6 layers of 32 tanh units, about 5,500 weights. It uses tanh because the momentum equation needs second derivatives of the output. The inputs are rescaled to the range -1 to 1 inside the network, so autograd still differentiates with respect to the physical coordinates. This is the first article in a series about PINNs for blood flow.

A from-scratch PyTorch build recovers blood flow, viscosity, and wall shear stress in a narrowed artery from 40 noisy velocity readings.

Wall shear stress is the friction blood puts on the wall of a vessel. It is tied to where plaque builds up, and it is hard to measure directly. A physics-informed neural network (PINN) is a natural fit. I built one in plain PyTorch, without DeepXDE or any other PINN library, for a 2D artery with a narrowing (a stenosis). From 40 velocity readings it reconstructs the velocity and pressure fields and the region of reversed flow behind the narrowing. It also works out the viscosity of the fluid, which I treated as unknown, and its wall shear stress follows the CFD reference closely.

The artery is a 2D channel of height H with a smooth bump on the lower wall that blocks half of the opening, a 50% stenosis. The flow is incompressible Navier-Stokes at a Reynolds number of 200. In these units the kinematic viscosity is 0.005. To score a reconstruction you need to know the right answer, so I generated it. A Navier-Stokes solver I wrote computes the steady flow, and the PINN only ever gets a few noisy readings taken from that solution. The noise is Gaussian, at 7% of the inlet velocity.

The network is a plain MLP with 6 layers of 32 tanh units, about 5,500 weights. It uses tanh because the momentum equation needs second derivatives of the output. The inputs are rescaled to the range -1 to 1 inside the network, so autograd still differentiates with respect to the physical coordinates. This is the first article in a series about PINNs for blood flow.

A from-scratch PyTorch build recovers blood flow, viscosity, and wall shear stress in a narrowed artery from 40 noisy velocity readings.

Wall shear stress is the friction blood puts on the wall of a vessel. It is tied to where plaque builds up, and it is hard to measure directly. A physics-informed neural network (PINN) is a natural fit. I built one in plain PyTorch, without DeepXDE or any other PINN library, for a 2D artery with a narrowing (a stenosis). From 40 velocity readings it reconstructs the velocity and pressure fields and the region of reversed flow behind the narrowing. It also works out the viscosity of the fluid, which I treated as unknown, and its wall shear stress follows the CFD reference closely.

The artery is a 2D channel of height H with a smooth bump on the lower wall that blocks half of the opening, a 50% stenosis. The flow is incompressible Navier-Stokes at a Reynolds number of 200. In these units the kinematic viscosity is 0.005. To score a reconstruction you need to know the right answer, so I generated it. A Navier-Stokes solver I wrote computes the steady flow, and the PINN only ever gets a few noisy readings taken from that solution. The noise is Gaussian, at 7% of the inlet velocity.

The network is a plain MLP with 6 layers of 32 tanh units, about 5,500 weights. It uses tanh because the momentum equation needs second derivatives of the output. The inputs are rescaled to the range -1 to 1 inside the network, so autograd still differentiates with respect to the physical coordinates. This is the first article in a series about PINNs for blood flow.

A from-scratch PyTorch build recovers blood flow, viscosity, and wall shear stress in a narrowed artery from 40 noisy velocity readings.

Wall shear stress is the friction blood puts on the wall of a vessel. It is tied to where plaque builds up, and it is hard to measure directly. A physics-informed neural network (PINN) is a natural fit. I built one in plain PyTorch, without DeepXDE or any other PINN library, for a 2D artery with a narrowing (a stenosis). From 40 velocity readings it reconstructs the velocity and pressure fields and the region of reversed flow behind the narrowing. It also works out the viscosity of the fluid, which I treated as unknown, and its wall shear stress follows the CFD reference closely.

The artery is a 2D channel of height H with a smooth bump on the lower wall that blocks half of the opening, a 50% stenosis. The flow is incompressible Navier-Stokes at a Reynolds number of 200. In these units the kinematic viscosity is 0.005. To score a reconstruction you need to know the right answer, so I generated it. A Navier-Stokes solver I wrote computes the steady flow, and the PINN only ever gets a few noisy readings taken from that solution. The noise is Gaussian, at 7% of the inlet velocity.

The network is a plain MLP with 6 layers of 32 tanh units, about 5,500 weights. It uses tanh because the momentum equation needs second derivatives of the output. The inputs are rescaled to the range -1 to 1 inside the network, so autograd still differentiates with respect to the physical coordinates. This is the first article in a series about PINNs for blood flow.

A from-scratch PyTorch build recovers blood flow, viscosity, and wall shear stress in a narrowed artery from 40 noisy velocity readings.

Wall shear stress is the friction blood puts on the wall of a vessel. It is tied to where plaque builds up, and it is hard to measure directly. A physics-informed neural network (PINN) is a natural fit. I built one in plain PyTorch, without DeepXDE or any other PINN library, for a 2D artery with a narrowing (a stenosis). From 40 velocity readings it reconstructs the velocity and pressure fields and the region of reversed flow behind the narrowing. It also works out the viscosity of the fluid, which I treated as unknown, and its wall shear stress follows the CFD reference closely.

The artery is a 2D channel of height H with a smooth bump on the lower wall that blocks half of the opening, a 50% stenosis. The flow is incompressible Navier-Stokes at a Reynolds number of 200. In these units the kinematic viscosity is 0.005. To score a reconstruction you need to know the right answer, so I generated it. A Navier-Stokes solver I wrote computes the steady flow, and the PINN only ever gets a few noisy readings taken from that solution. The noise is Gaussian, at 7% of the inlet velocity.

The network is a plain MLP with 6 layers of 32 tanh units, about 5,500 weights. It uses tanh because the momentum equation needs second derivatives of the output. The inputs are rescaled to the range -1 to 1 inside the network, so autograd still differentiates with respect to the physical coordinates. This is the first article in a series about PINNs for blood flow.

A from-scratch PyTorch build recovers blood flow, viscosity, and wall shear stress in a narrowed artery from 40 noisy velocity readings.

Wall shear stress is the friction blood puts on the wall of a vessel. It is tied to where plaque builds up, and it is hard to measure directly. A physics-informed neural network (PINN) is a natural fit. I built one in plain PyTorch, without DeepXDE or any other PINN library, for a 2D artery with a narrowing (a stenosis). From 40 velocity readings it reconstructs the velocity and pressure fields and the region of reversed flow behind the narrowing. It also works out the viscosity of the fluid, which I treated as unknown, and its wall shear stress follows the CFD reference closely.

The artery is a 2D channel of height H with a smooth bump on the lower wall that blocks half of the opening, a 50% stenosis. The flow is incompressible Navier-Stokes at a Reynolds number of 200. In these units the kinematic viscosity is 0.005. To score a reconstruction you need to know the right answer, so I generated it. A Navier-Stokes solver I wrote computes the steady flow, and the PINN only ever gets a few noisy readings taken from that solution. The noise is Gaussian, at 7% of the inlet velocity.

The network is a plain MLP with 6 layers of 32 tanh units, about 5,500 weights. It uses tanh because the momentum equation needs second derivatives of the output. The inputs are rescaled to the range -1 to 1 inside the network, so autograd still differentiates with respect to the physical coordinates. This is the first article in a series about PINNs for blood flow.

A from-scratch PyTorch build recovers blood flow, viscosity, and wall shear stress in a narrowed artery from 40 noisy velocity readings.

Wall shear stress is the friction blood puts on the wall of a vessel. It is tied to where plaque builds up, and it is hard to measure directly. A physics-informed neural network (PINN) is a natural fit. I built one in plain PyTorch, without DeepXDE or any other PINN library, for a 2D artery with a narrowing (a stenosis). From 40 velocity readings it reconstructs the velocity and pressure fields and the region of reversed flow behind the narrowing. It also works out the viscosity of the fluid, which I treated as unknown, and its wall shear stress follows the CFD reference closely.

The artery is a 2D channel of height H with a smooth bump on the lower wall that blocks half of the opening, a 50% stenosis. The flow is incompressible Navier-Stokes at a Reynolds number of 200. In these units the kinematic viscosity is 0.005. To score a reconstruction you need to know the right answer, so I generated it. A Navier-Stokes solver I wrote computes the steady flow, and the PINN only ever gets a few noisy readings taken from that solution. The noise is Gaussian, at 7% of the inlet velocity.

The network is a plain MLP with 6 layers of 32 tanh units, about 5,500 weights. It uses tanh because the momentum equation needs second derivatives of the output. The inputs are rescaled to the range -1 to 1 inside the network, so autograd still differentiates with respect to the physical coordinates. This is the first article in a series about PINNs for blood flow.

A from-scratch PyTorch build recovers blood flow, viscosity, and wall shear stress in a narrowed artery from 40 noisy velocity readings.

Wall shear stress is the friction blood puts on the wall of a vessel. It is tied to where plaque builds up, and it is hard to measure directly. A physics-informed neural network (PINN) is a natural fit. I built one in plain PyTorch, without DeepXDE or any other PINN library, for a 2D artery with a narrowing (a stenosis). From 40 velocity readings it reconstructs the velocity and pressure fields and the region of reversed flow behind the narrowing. It also works out the viscosity of the fluid, which I treated as unknown, and its wall shear stress follows the CFD reference closely.

The artery is a 2D channel of height H with a smooth bump on the lower wall that blocks half of the opening, a 50% stenosis. The flow is incompressible Navier-Stokes at a Reynolds number of 200. In these units the kinematic viscosity is 0.005. To score a reconstruction you need to know the right answer, so I generated it. A Navier-Stokes solver I wrote computes the steady flow, and the PINN only ever gets a few noisy readings taken from that solution. The noise is Gaussian, at 7% of the inlet velocity.

The network is a plain MLP with 6 layers of 32 tanh units, about 5,500 weights. It uses tanh because the momentum equation needs second derivatives of the output. The inputs are rescaled to the range -1 to 1 inside the network, so autograd still differentiates with respect to the physical coordinates. This is the first article in a series about PINNs for blood flow.

A from-scratch PyTorch build recovers blood flow, viscosity, and wall shear stress in a narrowed artery from 40 noisy velocity readings.

Wall shear stress is the friction blood puts on the wall of a vessel. It is tied to where plaque builds up, and it is hard to measure directly. A physics-informed neural network (PINN) is a natural fit. I built one in plain PyTorch, without DeepXDE or any other PINN library, for a 2D artery with a narrowing (a stenosis). From 40 velocity readings it reconstructs the velocity and pressure fields and the region of reversed flow behind the narrowing. It also works out the viscosity of the fluid, which I treated as unknown, and its wall shear stress follows the CFD reference closely.

The artery is a 2D channel of height H with a smooth bump on the lower wall that blocks half of the opening, a 50% stenosis. The flow is incompressible Navier-Stokes at a Reynolds number of 200. In these units the kinematic viscosity is 0.005. To score a reconstruction you need to know the right answer, so I generated it. A Navier-Stokes solver I wrote computes the steady flow, and the PINN only ever gets a few noisy readings taken from that solution. The noise is Gaussian, at 7% of the inlet velocity.

The network is a plain MLP with 6 layers of 32 tanh units, about 5,500 weights. It uses tanh because the momentum equation needs second derivatives of the output. The inputs are rescaled to the range -1 to 1 inside the network, so autograd still differentiates with respect to the physical coordinates. This is the first article in a series about PINNs for blood flow.

A from-scratch PyTorch build recovers blood flow, viscosity, and wall shear stress in a narrowed artery from 40 noisy velocity readings.

Wall shear stress is the friction blood puts on the wall of a vessel. It is tied to where plaque builds up, and it is hard to measure directly. A physics-informed neural network (PINN) is a natural fit. I built one in plain PyTorch, without DeepXDE or any other PINN library, for a 2D artery with a narrowing (a stenosis). From 40 velocity readings it reconstructs the velocity and pressure fields and the region of reversed flow behind the narrowing. It also works out the viscosity of the fluid, which I treated as unknown, and its wall shear stress follows the CFD reference closely.

The artery is a 2D channel of height H with a smooth bump on the lower wall that blocks half of the opening, a 50% stenosis. The flow is incompressible Navier-Stokes at a Reynolds number of 200. In these units the kinematic viscosity is 0.005. To score a reconstruction you need to know the right answer, so I generated it. A Navier-Stokes solver I wrote computes the steady flow, and the PINN only ever gets a few noisy readings taken from that solution. The noise is Gaussian, at 7% of the inlet velocity.

The network is a plain MLP with 6 layers of 32 tanh units, about 5,500 weights. It uses tanh because the momentum equation needs second derivatives of the output. The inputs are rescaled to the range -1 to 1 inside the network, so autograd still differentiates with respect to the physical coordinates. This is the first article in a series about PINNs for blood flow.

A from-scratch PyTorch build recovers blood flow, viscosity, and wall shear stress in a narrowed artery from 40 noisy velocity readings.

Wall shear stress is the friction blood puts on the wall of a vessel. It is tied to where plaque builds up, and it is hard to measure directly. A physics-informed neural network (PINN) is a natural fit. I built one in plain PyTorch, without DeepXDE or any other PINN library, for a 2D artery with a narrowing (a stenosis). From 40 velocity readings it reconstructs the velocity and pressure fields and the region of reversed flow behind the narrowing. It also works out the viscosity of the fluid, which I treated as unknown, and its wall shear stress follows the CFD reference closely.

The artery is a 2D channel of height H with a smooth bump on the lower wall that blocks half of the opening, a 50% stenosis. The flow is incompressible Navier-Stokes at a Reynolds number of 200. In these units the kinematic viscosity is 0.005. To score a reconstruction you need to know the right answer, so I generated it. A Navier-Stokes solver I wrote computes the steady flow, and the PINN only ever gets a few noisy readings taken from that solution. The noise is Gaussian, at 7% of the inlet velocity.

The network is a plain MLP with 6 layers of 32 tanh units, about 5,500 weights. It uses tanh because the momentum equation needs second derivatives of the output. The inputs are rescaled to the range -1 to 1 inside the network, so autograd still differentiates with respect to the physical coordinates. This is the first article in a series about PINNs for blood flow.

A from-scratch PyTorch build recovers blood flow, viscosity, and wall shear stress in a narrowed artery from 40 noisy velocity readings.

Wall shear stress is the friction blood puts on the wall of a vessel. It is tied to where plaque builds up, and it is hard to measure directly. A physics-informed neural network (PINN) is a natural fit. I built one in plain PyTorch, without DeepXDE or any other PINN library, for a 2D artery with a narrowing (a stenosis). From 40 velocity readings it reconstructs the velocity and pressure fields and the region of reversed flow behind the narrowing. It also works out the viscosity of the fluid, which I treated as unknown, and its wall shear stress follows the CFD reference closely.

The artery is a 2D channel of height H with a smooth bump on the lower wall that blocks half of the opening, a 50% stenosis. The flow is incompressible Navier-Stokes at a Reynolds number of 200. In these units the kinematic viscosity is 0.005. To score a reconstruction you need to know the right answer, so I generated it. A Navier-Stokes solver I wrote computes the steady flow, and the PINN only ever gets a few noisy readings taken from that solution. The noise is Gaussian, at 7% of the inlet velocity.

The network is a plain MLP with 6 layers of 32 tanh units, about 5,500 weights. It uses tanh because the momentum equation needs second derivatives of the output. The inputs are rescaled to the range -1 to 1 inside the network, so autograd still differentiates with respect to the physical coordinates. This is the first article in a series about PINNs for blood flow.