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Hierarchical Implicit Neural Emulators
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Proposes a multiscale implicit neural emulator that conditions next-step prediction on a hierarchy of downsampled future-state latents, evaluated on 2D turbulent Navier-Stokes with long rollouts.
Problem
Autoregressive neural PDE emulators accumulate error and become unstable over long horizons for multiscale chaotic systems such as turbulent flows.
Approach
Augments the autoregressive input with coarse-grained (spatially downsampled) latent representations of several future states at increasing compression rates, mimicking iterative refinement of implicit time-stepping methods without iterative solves; supervision on both the physical state and future latents is applied during training.
Key results
- Ours L=3 achieves rollout MSE 3.37e-02 at step 25 vs 8.04e-02 for the 1-step baseline (L=1) on 256x256 jet data, and reports over 50% MSE reduction at steps 25-50.
- Over 2x10^5 steps (10x training length), the L=3 model achieves a 93.0% stability rate vs 38.0% for the next best baseline (2-step History).
- L=3 adds only ~1.8M parameters (36.67M vs 34.88M) with forward time 0.031s vs 0.030s for the baseline.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the three-level future-state hierarchy instead of the one-step baseline | UNet with Fourier layers · 2D Navier-Stokes, Re=10^4, 256×256; 25–100-step rollout | MSE: 3.37e-02 (1.41e-02) vs 8.04e-02 (3.45e-02) at 25 steps; 5.92e-01 (1.84e-01) vs 9.38e-01 (2.88e-01) at 100 steps | Helped |
| use the three-level future-state hierarchy instead of 3-step History | UNet with Fourier layers · 2D Navier-Stokes, Re=10^4, 256×256; 25–100-step rollout | MSE: 3.37e-02 (1.41e-02) vs 4.28e-02 (2.10e-02) at 25 steps; 5.92e-01 (1.84e-01) vs 6.81e-01 (2.39e-01) at 100 steps | Helped |
| use the three-level future-state hierarchy instead of 2-Step History | UNet with Fourier layers · 2D Navier-Stokes, Re=10^4; 2×10^5-step rollout | stability rate: 93.0% vs 38.0% | Helped |
| add the three-level future-state hierarchy to the L=1 baseline | UNet with Fourier layers · 2D Navier-Stokes, Re=10^4, 256×256 | forward time: 0.030 -> 0.031 | Hurt |
| set downsampling ratio r¹=2 rather than r¹=8 (ablation) | L=2 emulator · 2D Navier-Stokes, Re=10^4, 256×256; 50-step rollout | MSE: 1.606e-01 (6.496e-02) vs 3.774e-01 (1.797e-01) | Hurt |
| set downsampling ratio r¹=4 rather than r¹=8 (ablation) | L=2 emulator · 2D Navier-Stokes, Re=10^4, 256×256; 50-step rollout | MSE: 1.606e-01 (6.496e-02) vs 1.998e-01 (1.074e-01) | Hurt |
| set downsampling ratio r¹=16 rather than r¹=8 (ablation) | L=2 emulator · 2D Navier-Stokes, Re=10^4, 256×256; 50-step rollout | MSE: 1.606e-01 (6.496e-02) vs 2.039e-01 (8.009e-02) | Hurt |
| increase hierarchy depth from L=1 to L=3 (ablation) | Implicit neural emulator · 2D Navier-Stokes turbulence | temporal structure and high-frequency accuracy | Helped |
Surprising findings
- Basic hierarchical baselines using spatial or history hierarchies provide minimal gains over the 1-step baseline, suggesting temporal hierarchy of future states is the key factor.
- A 4x larger FNO baseline does not match the proposed model's stability and accuracy on the turbulence benchmark.
Limitations
- Focused on deterministic modeling; probabilistic forecasting not addressed.
- Evaluated only on 2D turbulent vorticity flows with a single simulator and limited configurations.
- Stability criterion is a binary energy-based threshold, which may not capture all physical consistency.
Experimental setup
Datasets: 2D Navier-Stokes vorticity simulations generated with py2d (Re=10^4, beta=20; also Re=5x10^3 and beta=0 eddy configurati
Benchmarks: 2D Navier-Stokes turbulence (Re=10^4 with zonal jets) 2D Navier-Stokes eddy configuration (beta=0)
Metrics: MSE energy-based stability rate energy spectrum error zonal mean parameter count forward time
Baselines: 1-step autoregressive (L=1) 2-step ahead 2-step history 3-step history spatial hierarchy history hierarchy FNO (4x params)
Builds on: U-Net Fourier neural operator Adams-Moulton multistep methods pushforward training
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