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Riemann Tensor Neural Networks: Learning Conservative Systems with Physics-Constrained Networks
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Proposes Riemann Tensor Neural Networks (RTNNs), an MLP-based architecture that outputs divergence-free symmetric tensors by construction, and uses it as a PINN-style surrogate for Euler, Navier-Stokes, and MHD flows.
Problem
Enforcing conservation laws (mass and momentum) in neural PDE surrogates; soft PINN penalties do not guarantee divergence-free structure and optimization is stiff.
Approach
Parametrize a symmetric tensor field as a sum of fixed Riemann-like (0,4)-tensor basis terms contracted with the second derivatives of learned scalar potentials, so the divergence-free condition holds identically; the flux tensor encodes density, momentum, and stress for conservative PDEs.
Key results
- On the Euler vortex, RTNN reaches median relative L2 error 9.92e-05 versus 3.82e-02 for PINN and 3.87e-01 for NCL.
- On the cylinder wake, RTNN achieves 5.70e-03 relative L2 error versus 2.99e-02 (PINN) and 2.54e-02 (NCL); on Beltrami 4.28e-04 versus 1.41e-03 (PINN).
- On 2D MHD, RTNN reduces velocity relative L2 error to 2.34e-02 from 1.61e-01 (Curl-SPINN) and 1.82e-01 (SPINN), at about 2x the wall time of SPINN.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use RTNN instead of PINN | Euler vortex surrogate · 2D isentropic Euler vortex | relative L2 error: 3.82e-02 -> 9.92e-05 | Helped |
| use RTNN instead of PINN | incompressible Navier-Stokes surrogate · Cylinder wake | rL2 error: 2.99e-02 -> 5.70e-03 | Helped |
| use RTNN instead of PINN | incompressible Navier-Stokes surrogate · NACA 0012 airfoil flow | rL2 error: 2.48e-01 -> 1.44e-02 | Helped |
| use RTNN instead of PINN | incompressible Navier-Stokes surrogate · 3D Beltrami flow | rL2 error: 1.41e-03 -> 4.28e-04 | Helped |
| use RTNN instead of Curl-SPINN | MHD surrogate · 3D periodic incompressible MHD | velocity rL2 error: 1.61e-01 -> 2.34e-02 | Helped |
| use RTNN instead of Curl-SPINN | MHD surrogate · 3D periodic incompressible MHD | magnetic-field rL2 error: 1.55e-01 -> 1.04e-01 | Helped |
| use RTNN instead of PINN | MHD surrogate · 3D periodic incompressible MHD | wall time (s): 414.73 -> 2340.41 | Hurt |
| use RTNN instead of PINN | Euler vortex surrogate · 2D isentropic Euler vortex | wall time (s): 365.67 -> 596.34 | Hurt |
| use RTNN instead of PINN | incompressible Navier-Stokes surrogate · Cylinder wake | time (s): 3.12e+02 -> 1.21e+03 | Hurt |
| use RTNN instead of PINN | incompressible Navier-Stokes surrogate · NACA 0012 airfoil flow | time (s): 1.06e+03 -> 1.10e+03 | Mixed |
| use RTNN instead of PINN | incompressible Navier-Stokes surrogate · 3D Beltrami flow | time (s): 1.82e+02 -> 2.97e+02 | Hurt |
| use the extended momentum equation instead of the stress-tensor loss (ablation) | RTNN · 3D Beltrami flow | L2 median: 4.28e-04 -> 4.55e-04 | Hurt |
Surprising findings
- Wall-time comparisons show RTNN is often slower than PINN and NCL, with PINN faster on the cylinder and airfoil cases (e.g., 312 s vs 1210 s on cylinder).
Limitations
- Experiments are limited to fluid and MHD problems with small MLPs and collocation-based PINN-style training rather than operator learning.
- Requires fourth-order derivatives, increasing computational cost.
- Velocity recovery divides by density, causing instability when density is small; an identity offset is used as a workaround.
Experimental setup
Datasets: analytical Euler vortex solution analytical Beltrami flow DNS cylinder wake data in-house RANS solver data (NACA 0012) spectral solver MHD data
Benchmarks: 2D isentropic Euler vortex 3D Beltrami flow 2D cylinder wake (Re=100) steady NACA 0012 airfoil (Re=1000) 3D periodic incompressible MHD
Metrics: median relative L2 error wall time
Baselines: PINN NCL SPINN Curl-SPINN
Builds on: physics-informed neural networks neural conservation laws separable physics-informed neural networks taylor-mode automatic differentiation
Closest papers by meaning
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