Research

Can a neural network solve the Navier–Stokes equations as well as a classical method — and how do we know when to trust the answer?

Overview

My PhD research develops physics-informed neural networks (PINNs) as an alternative — and a complement — to classical mesh-based solvers for the incompressible Navier–Stokes system. The work spans three threads: rigorous benchmarking against analytic solutions, hybrid PINN/finite-volume formulations, and translational applications in medical imaging and material science.

The mathematical setting

The core problem is the incompressible Navier–Stokes system on a domain \Omega \subset \mathbb{R}^3 with t \in [0, T]:

\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} \;=\; -\frac{1}{\rho}\nabla p + \nu\, \Delta \mathbf{u} + \mathbf{f}, \qquad \nabla \cdot \mathbf{u} = 0,

where \mathbf{u}(x,y,z,t) = (u, v, w)^\top is the velocity, p is the pressure, \rho the density, \nu the kinematic viscosity, and \mathbf{f} a body force.

A PINN \mathcal{N}_\theta : \mathbb{R}^4 \to \mathbb{R}^4 is trained by minimizing a composite loss:

\mathcal{L}(\theta) = \lambda_{\text{pde}}\, \mathcal{L}_{\text{pde}} + \lambda_{\text{div}}\, \mathcal{L}_{\nabla\cdot\mathbf{u}} + \lambda_{\text{bc}}\, \mathcal{L}_{\text{bc}} + \lambda_{\text{ic}}\, \mathcal{L}_{\text{ic}} + \lambda_{\text{data}}\, \mathcal{L}_{\text{data}},

with each term an L^2 residual on collocation points sampled by Sobol or residual-adaptive strategies. Differential operators are realized via JAX / PyTorch automatic differentiation.

Research threads

1. Benchmarking PINNs against analytic solutions

Every PINN result is reported alongside a classical baseline (projection method, spectral solver, RK4) on canonical flows where ground truth is known: Taylor–Green vortex, Kovasznay flow, and Beltrami flow (Ethier & Steinman 1994).

Reported metrics: L^2 relative error, L^\infty error, divergence \lVert \nabla \cdot \mathbf{u} \rVert_2, wall-clock, and GPU memory.

2. Hybrid PINN / finite-volume formulations

I am exploring formulations that combine PINN flexibility (mesh-free, easily incorporates data) with the conservation guarantees of finite-volume schemes — trying to inherit the best of both worlds while keeping training stable on high-Reynolds-number flows.

3. Cerebrospinal fluid flow in Parkinson’s disease

The longer-term application is inverse problems on 4D-flow MRI of cerebrospinal fluid (CSF). The goal: recover effective viscosity and flow parameters from sparse, noisy clinical data, where classical solvers struggle with the lack of clean boundary conditions but PINNs can ingest the data directly.

4. Material science applications

Secondary application area: pore-scale flow simulation in heterogeneous porous media, where multi-scale PINN architectures show promise.

Tooling and reproducibility

  • Compute: NVIDIA A100 / H100; multi-GPU via jax.pmap or torch.distributed.
  • Stack: JAX + Equinox, PyTorch, Hydra (config), Weights & Biases (logging), DeepXDE / NVIDIA Modulus (reference baselines).
  • Numerical baselines: FEniCSx, Dedalus (spectral), in-house projection solver.
  • Every experiment writes a manifest.json with git SHA, config hash, seed, hardware, library versions, wall-clock, and final metrics — fully reproducible from a single config file.

Current focus

Validating a 3D Navier–Stokes PINN against the Beltrami benchmark, with a side-by-side classical projection-method comparison ready for the next group meeting.