Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Latent Mamba Operator for Partial Differential Equations
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Proposes LaMO, a latent-space bidirectional state-space-model neural operator for PDEs that is theoretically linked to kernel integral operators and reports large error reductions on Darcy, Navier-Stokes, Plasticity, Airfoil, Pipe and Elasticity benchmarks.
Problem
Existing neural operators for PDEs struggle with scalability to high-dimensional and high-resolution domains, high computational cost (quadratic attention), and capturing continuous long-range dependencies in PDE dynamics.
Approach
Compress physical tokens into a small set of adaptive latent tokens (softmax-weighted, Perceiver-style) and apply bidirectional Mamba-style selective SSM blocks in latent space, with a theorem that the latent SSM acts as a learnable kernel integral.
Key results
- LaMO achieves mean relative L2 error of 0.0050 on Elasticity, 0.0460 on Navier-Stokes, 0.0039 on Darcy, 0.0007 on Plasticity, 0.0041 on Airfoil and 0.0038 on Pipe, best on all six benchmarks in the main table.
- The paper reports an average 32.3% error reduction over the second-best baseline (Transolver) across benchmarks.
- LaMO reportedly uses 3x fewer parameters and runs 1.8x faster than transformer-based baselines on Darcy and Navier-Stokes.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace Transolver with LaMO | PDE operator learning · Elasticity | mean relative ℓ2 error: 0.0064 -> 0.0050 | Helped |
| replace Transolver with LaMO | PDE operator learning · Navier–Stokes | mean relative ℓ2 error: 0.0957 -> 0.0460 | Helped |
| replace Transolver with LaMO | PDE operator learning · Darcy | mean relative ℓ2 error: 0.0059 -> 0.0039 | Helped |
| replace Transolver with LaMO | PDE operator learning · Plasticity | mean relative ℓ2 error: 0.0013 -> 0.0007 | Helped |
| replace Transolver with LaMO | PDE operator learning · Airfoil | mean relative ℓ2 error: 0.0053 -> 0.0041 | Helped |
| replace Transolver with LaMO | PDE operator learning · Pipe | mean relative ℓ2 error: 0.0046 -> 0.0038 | Helped |
| increase the number of latent tokens (ablation) | LaMO · Airfoil | relative L2: 0.0079 -> 0.0039 -> 0.0048 | Mixed |
| increase SSM DState with 1936 latent tokens and expand dim 1 (ablation) | LaMO · Darcy | relative L2 loss: 0.0045 -> 0.0040 | Helped |
| increase SSM DState with 121 latent tokens and expand dim 1 (ablation) | LaMO · Darcy | relative L2 loss: 0.0068 -> 0.0071 | Mixed |
| increase latent tokens from 121 to 1936 with expand dim 2 and DState 64 (ablation) | LaMO · Darcy | relative L2 loss: 0.0066 -> 0.0038 | Helped |
| use unidirectional instead of bidirectional SSM (ablation) | LaMO · Plasticity | relative ℓ2 error: 0.0007 -> 0.0022 | Hurt |
| use bidirectional SSM instead of patches (ablation) | LaMO · Darcy | relative ℓ2 error: 0.0039 -> 0.0050 | Hurt |
Surprising findings
- Unidirectional SSM performs inconsistently; on Plasticity its error rises from 0.0007 to 0.0022 relative to bidirectional.
Limitations
- Pretraining and foundation-model use of the SSM backbone remain unexplored.
- Latent-token layout relies on patch-like grids for regular domains; regular-grid Perceiver-style latent tokens scale poorly with resolution per the authors.
- Theoretical connection to kernel integrals relies on approximations (Monte Carlo, Euler/ZOH, determinant set to 1).
Experimental setup
Datasets: Darcy Navier-Stokes Plasticity Airfoil Pipe Elasticity
Benchmarks: Darcy Navier-Stokes Plasticity Airfoil Pipe Elasticity
Metrics: mean relative L2 error
Baselines: UNET ResNet SWIN DeepONet WMT U-FNO FNO U-NO F-FNO LSM Galerkin Transformer HT-Net
Builds on: Mamba Transolver Fourier neural operator Perceiver Structured state space (S4)
Closest papers by meaning
- Alias-Free Mamba Neural Operator (NeurIPS 2024 · similarity 0.91)
- Latent Neural Operator for Solving Forward and Inverse PDE Problems (NeurIPS 2024 · similarity 0.88)
- BENO: Boundary-embedded Neural Operators for Elliptic PDEs (ICLR 2024 · similarity 0.88)
- Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equations (ICLR 2025 · similarity 0.86)
- Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEs (ICML 2024 · similarity 0.86)
- Component Fourier Neural Operator for Singularly Perturbed Differential Equations (AAAI 2024 · similarity 0.86)
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