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Space-Time Continuous PDE Forecasting using Equivariant Neural Fields
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Proposes a space-time continuous PDE solver that models solutions as equivariant latent-space flows of an SE(n)-style equivariant neural field, with meta-learned initial latents, and evaluates on heat, Navier-Stokes, shallow-water and convection PDEs.
Problem
Neural-field-based PDE forecasting ignores known PDE symmetries and boundary conditions, so it generalizes poorly to unseen geometric transformations of initial conditions and to complex geometries such as the sphere, torus and ball.
Approach
Extends equivariant neural fields (ENFs) to PDE-specific symmetry groups (torus translations, S^2 rotations about an axis, SO(3), ball) via custom bi-invariant attributes, models PDE solutions as neural-ODE flows of pose-context latents using log/exp-map message passing, and uses meta-learned initial latents for fast inference.
Key results
- On Navier-Stokes T2 with 5% observed initial condition, the equivariant model (a^T2) reaches t_in test MSE 3.44E-2 vs 3.94E-2 for DINo and 3.40E-1 for G-FNO.
- On the 3D ball convection task, the equivariant model reaches t_in test MSE 5.99E-4 vs 3.06E-3 for DINo.
- On the planar heat equation with disjoint train/test spike locations, the non-equivariant model and CNODE fail on test initial conditions whereas the SE(2)-equivariant model performs well.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add periodic-translation equivariance (ablation) | Ours on Navier-Stokes on 𝕋² · Navier-Stokes on 𝕋², 100% observed, t_in test | MSE: 2.60{ ±3.16}E-02 -> 1.57{ ±0.09}E-03 | Helped |
| Add meta-learning instead of autodecoding (ablation) | Ours with 𝕋²/π equivariance on Navier-Stokes · Navier-Stokes on 𝕋², 100% observed, t_in test | MSE: 6.75{ ±0.62}E-02 -> 1.57{ ±0.09}E-03 | Helped |
| Add periodic-translation equivariance (ablation) | Ours on Navier-Stokes on 𝕋² · Navier-Stokes on 𝕋², 50% observed, t_in test | MSE: 2.60{ ±3.16}E-02 -> 5.75{ ±2.58}E-03 | Helped |
| Add periodic-translation equivariance (ablation) | Ours on Navier-Stokes on 𝕋² · Navier-Stokes on 𝕋², 5% observed, t_in test | MSE: 7.96{ ±1.65}E-02 -> 3.44{ ±1.43}E-02 | Helped |
| Add meta-learning instead of autodecoding (ablation) | Ours with 𝕋²/π equivariance on Navier-Stokes · Navier-Stokes on 𝕋², 50% observed, t_in test | MSE: 7.01{ ±3.56}E-02 -> 5.75{ ±2.58}E-03 | Helped |
| Add meta-learning instead of autodecoding (ablation) | Ours with 𝕋²/π equivariance on Navier-Stokes · Navier-Stokes on 𝕋², 5% observed, t_in test | MSE: 7.01{ ±3.56}E-02 -> 3.44{ ±1.43}E-02 | Helped |
| Use the proposed continuous equivariant solver rather than DINo | Shallow-water on S² · Shallow-water on S², zero-shot 2× super-resolution, t_in test | MSE: 3.37E-04 -> 1.61 { ±0.01}E-04 | Helped |
| Use the proposed continuous equivariant solver rather than DINo | Internally-heated convection in 𝔹³ · Internally-heated convection in 𝔹³, t_in test | MSE: 3.06E-03 -> 5.99{ ±0.15}E-04 | Helped |
| Use the proposed continuous equivariant solver rather than DINo | Shallow-water on S² · Shallow-water on S², train resolution, t_out test | MSE: 1.37E-03 -> 1.87{ ±0.04}E-03 | Hurt |
Surprising findings
- On fully observed Navier-Stokes T2, regular-grid baselines G-FNO and FNO outperform the proposed continuous method, which is competitive but not best.
- Equivariant model's error accumulates beyond the training horizon on shallow-water, failing to capture dynamics for t > 9.
Limitations
- Error accumulation over long horizons beyond the training window, especially on shallow-water.
- Global attention in the ENF backbone incurs steep computational cost when increasing capacity to reduce reconstruction error.
- Fully observed regular-grid baselines (G-FNO, FNO) remain stronger on Navier-Stokes T2 in the dense setting.
Experimental setup
Datasets: Planar heat diffusion (py-pde) Navier-Stokes on flat 2-torus (py-pde, Gaussian random field ICs) Heat diffusion on S^2 (Dedalus) Global shallow-water equations on S^2 (Dedalus) Internally-heated convection in 3D ball (Dedalus)
Benchmarks: Planar heat diffusion equivariance test Navier-Stokes on T2 Heat diffusion on sphere (data efficiency) Shallow-water on sphere (super-resolution) Internally-heated convection in ball
Metrics: MSE
Baselines: CNODE FNO G-FNO DINo
Builds on: Conditional neural fields for PDE modelling (CNF-ODE / DINo) Equivariant Neural Fields PONITA Meta-SGD Neural ODE
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