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README.md

@@ -1,50 +1,200 @@
 ---
 license: mit
 library_name: pytorch
-tags: [neural-operator, pde-solver, transolver, linear-attention, elasticity]
-metrics: [relative-l2]
+tags:
+- neural-operator
+- pde-solver
+- physics-informed
+- elasticity
+- linear-attention
+- transolver
+metrics:
+- relative-l2
 ---
 
-# LinearNO — Elastic Stress Surrogate (Geo-FNO Elasticity)
+# Model Card for Equilibrium-Regularized LinearNO — Elastic Stress Surrogate (Geo-FNO Elasticity)
 
-Predicts the per-node **von Mises stress** of a hyper-elastic unit cell with a central void on
-the Geo-FNO **Elasticity** benchmark. Attention block: **LinearNO** (asymmetric linear attention,
-`shared_qk`, `project_out=False`), a drop-in
-replacement for Transolver's Physics-Attention.
+A neural-operator surrogate that predicts the per-node von Mises stress field of a hyper-elastic
+unit cell with a central void, on the **Geo-FNO Elasticity** benchmark. It maps a 2-D unstructured
+mesh (972 nodes) directly to the stress field in milliseconds on CPU, replacing a per-geometry
+finite-element (FEM) solve for fast design-space exploration.
 
-- **Test relative L2:** 0.005968 (this checkpoint).
-- **Params:** 582275 (≤ the Transolver baseline, 713,665).
-- **Interactive demo:** https://huggingface.co/spaces/Efradeca/elastic-stress-surrogate
+## Model Details
 
-## Results (Geo-FNO Elasticity test relative L2, mean ± std over 3 seeds)
+### Model Description
 
-| Model | test rel-L2 | Params | Notes |
-|---|---|---|---|
-| Transolver baseline (reproduced) | 0.00678 ± 0.0012 | 713,665 | reproduces the published 0.0064 |
-| LinearNO | 0.00664 ± 0.0008 | 713,089 | ≤ baseline params; does **not** reach the paper's 0.0050 |
-| **LinearNO + equilibrium reg (this model)** | 0.00668 ± 0.0006 | 582,275 | accuracy preserved + **~360× lower ∇·σ residual** |
+The model is a Transolver-family transformer operator whose attention block is the asymmetric
+linear-attention operator **LinearNO**. Its distinguishing component is an **equilibrium-residual
+regularizer**: the network predicts the full stress tensor (σ_xx, σ_yy, σ_xy) and is penalized by a
+discrete divergence operator so the predicted field approaches static mechanical equilibrium
+(∇·σ ≈ 0), yielding physically consistent predictions at no measured accuracy cost.
+
+- **Developed by:** Efradeca
+- **Model type:** Neural operator (transformer PDE surrogate) for 2-D static hyper-elastic stress
+- **Inputs / outputs:** node coordinates (B, 972, 2) → per-node von Mises stress (B, 972)
+- **License:** MIT
+- **Built on (not original to this work):** the **Transolver** solver (Wu et al., ICML 2024) and a
+  reimplementation of the **LinearNO** attention block (Hu et al., AAAI 2026). The original
+  contribution here is the equilibrium-residual regularizer and the out-of-distribution analysis.
+
+### Model Sources
+
+- **Papers:** Transolver (arXiv:2402.02366); LinearNO (arXiv:2511.06294); Geo-FNO dataset (arXiv:2207.05209)
+- **Demo:** https://huggingface.co/spaces/Efradeca/elastic-stress-surrogate
+
+## Uses
+
+### Direct Use
+
+Millisecond-scale, CPU prediction of the von Mises stress field of a hyper-elastic unit cell with a
+central void, for design-space exploration / shape optimization of the void (screen many candidate
+geometries, then verify the few best with FEM).
+
+### Downstream Use
+
+A starting point for fine-tuning to related geometries, materials, or boundary conditions, which
+requires retraining on the corresponding FEM data.
+
+### Out-of-Scope Use
+
+This is a narrow surrogate, not a general stress solver. It is **not** valid, without retraining,
+for other materials (e.g. steel), other boundary conditions, or geometries outside the training
+distribution, and it must not be used as a certified solver for safety-critical decisions.
+
+## Bias, Risks, and Limitations
 
-The equilibrium regularizer reduces the discrete equilibrium residual ‖∇·σ‖² by ~360× (6.1e6 → 1.7e4)
-with no accuracy cost, and this physical consistency is **maintained out-of-distribution** (OOD residual
-1.7e4 vs in-dist 1.6e4). It does **not** improve OOD *accuracy* (degradation comparable to LinearNO) —
-reported honestly, not massaged.
+- **Narrow domain:** a single benchmark, one hyper-elastic (rubber-like) material, one geometry
+  family (unit cell, central void, radius 0.2–0.4, clamped bottom, tensile top).
+- **Run-to-run variance:** the training set is small (1000 samples), so per-seed results vary
+  noticeably; results are reported as mean ± std with the full per-seed distribution (see Evaluation).
+- **Reimplemented component:** LinearNO has no official public reference implementation; it was
+  reproduced from the paper's equations.
+- **Regularizer scope:** the supervised target is scalar von Mises stress; the tensor components are
+  latent and shaped by a *discrete* ∇·σ penalty (validated against analytic fields), which is a
+  physical-consistency prior, not exact continuous momentum balance.
+- **Out-of-distribution:** accuracy degrades on geometries far from the training distribution
+  (extreme voids); the regularizer maintains physical consistency OOD but does not improve OOD accuracy.
+
+### Recommendations
+
+Use within the training distribution; verify any safety-critical prediction with FEM; monitor the
+equilibrium residual ‖∇·σ‖² as a physical-consistency indicator.
+
+## How to Get Started with the Model
 
-## Usage
 ```python
-import torch, json
-from safetensors.torch import load_file
-from model import load_checkpoint  # bundled loader (or use stress_operator.models.build_model)
+from huggingface_hub import snapshot_download
+from model import load_checkpoint, predict_stress   # bundled with the repo
+
+repo = snapshot_download("Efradeca/transolver-linearno-elasticity")
+model, normalizer, info = load_checkpoint(f"{repo}/model.safetensors", device="cpu")
+
+# coords: (N, 2) node coordinates of a unit cell with a central void
+stress = predict_stress(model, coords, normalizer, info)   # (N,) von Mises stress
 ```
 
-## Honesty caveats (read these)
-1. **Scope:** a *narrow* result on one benchmark — **not** a foundation model. The material is
-   hyper-elastic (rubber-like), **not steel**; the method transfers, a steel result needs steel data.
-2. **LinearNO** has no official public repo; it was reimplemented from the paper's equations
-   (arXiv:2511.06294). Only the attention block differs from Transolver (ICML 2024, arXiv:2402.02366, MIT).
-3. **Reproduction variance:** small dataset (1000 train) → run-to-run variance; numbers are
-   eager (deterministic), mean ± std over seeds {0,1,2}.
-4. The interactive demo uses a **synthetic** mesh (no FEM ground truth for arbitrary void shapes).
+## Training Details
+
+### Training Data
+
+Geo-FNO Elasticity (Li et al., 2022): FEM simulations of a hyper-elastic unit cell with a random
+central void (radius 0.2–0.4), 972 nodes per sample, per-node von Mises stress. Split: 1000 train /
+200 test (first-1000 / last-200 of 2000, following the upstream Transolver protocol).
+
+### Training Procedure
+
+#### Preprocessing
+
+Global z-score normalization of the stress target (de-normalized before the metric). The
+equilibrium-regularized model outputs three stress-tensor channels and derives von Mises.
+
+#### Training Hyperparameters
+
+- **Architecture:** 8 layers, hidden dim 128, 8 heads, dim_head 16, slices M = 64;
+  attention `linearno` (variant `shared_qk`, project_out=`False`)
+- **Optimization:** AdamW, lr 1e-3, weight decay 1e-5, cosine annealing, 500 epochs, batch size 1,
+  gradient clipping 0.1
+- **Loss:** relative L2 + λ·‖∇·σ‖² on interior nodes (λ = 0.01)
+- **Seeds:** {0, 1, 2} (mean ± std reported)
+
+## Evaluation
+
+### Testing Data, Factors & Metrics
+
+- **Testing data:** the 200 held-out FEM meshes.
+- **Factors:** in-distribution vs. out-of-distribution (geometry-stratified by void size).
+- **Metrics:** relative L2 (primary); the discrete equilibrium residual ‖∇·σ‖²; per-node Pearson r
+  and R² against the FEM ground truth; peak-stress relative error.
+
+### Results
+
+Geo-FNO Elasticity test relative L2 (mean ± std over seeds; per-seed values shown because variance
+is large and is not hidden):
+
+| Model | mean ± std | per-seed | params |
+|---|---|---|---|
+| Published Transolver (Wu et al. 2024) | 0.0064 | — | ~0.7M |
+| Published LinearNO (Hu et al. 2026, M=64) | 0.0050 | — | — |
+| Transolver baseline (this work, reproduced) | 0.00678 ± 0.0012 | 0.00587 / 0.00606 / 0.00841 | 713,665 |
+| LinearNO (this work, reimplemented) | 0.00741 ± 0.0025 | 0.00592 / 0.00545 / 0.01086 | 713,089 |
+| **+ equilibrium regularizer (this model)** | 0.00668 ± 0.0006 | 0.00597 / 0.00653 / 0.00754 | 582,275 |
+
+**Physical consistency.** The regularizer reduces the discrete equilibrium residual ‖∇·σ‖² by
+**~360×** (6.1e6 → 1.7e4) at no measured accuracy cost, and this consistency is maintained
+out-of-distribution (OOD residual 1.73e4 vs. in-distribution 1.61e4).
+
+**Verification against FEM ground truth** (200 test meshes): per-sample relative L2 mean 0.00597
+(median 0.00533), pooled Pearson r = 0.9999, R² = 0.9999, peak-stress relative error 0.3%.
+
+#### Summary
+
+The reproduced Transolver baseline matches the published 0.0064 within ~6%. On accuracy, LinearNO and
+the regularized model are statistically comparable to the baseline at ≤ its parameter count; we do
+not claim an accuracy improvement (two of three LinearNO seeds reach the published level while one
+seed lands in a degenerate generalization basin, inflating the 3-seed mean). The contribution is
+physical consistency: a stress field that satisfies discrete static equilibrium, preserved under
+covariate shift, at no accuracy cost and with fewer parameters than the baseline.
+
+## Technical Specifications
+
+### Model Architecture and Objective
+
+Transolver-family transformer (encoder → 8 pre-norm blocks → linear decoder head) with the LinearNO
+asymmetric linear-attention block; objective = relative-L2 data loss + equilibrium-residual penalty.
+
+### Compute Infrastructure
+
+- **Hardware:** single NVIDIA A10 GPU (Modal).
+- **Software:** PyTorch, einops, safetensors; CPU inference for the demo.
 
 ## Citation
-Transolver (Wu et al., ICML 2024, arXiv:2402.02366) · LinearNO (Hu et al., AAAI 2026,
-arXiv:2511.06294) · Geo-FNO (Li et al., 2022, arXiv:2207.05209).
+
+**BibTeX:**
+```bibtex
+@inproceedings{wu2024transolver,
+  title={Transolver: A Fast Transformer Solver for PDEs on General Geometries},
+  author={Wu, Haixu and Luo, Huakun and Wang, Haowen and Wang, Jianmin and Long, Mingsheng},
+  booktitle={International Conference on Machine Learning (ICML)},
+  year={2024}
+}
+@inproceedings{hu2026linearno,
+  title={Transolver is a Linear Transformer: Revisiting Physics-Attention through the Lens of Linear Attention},
+  author={Hu and Liu and Qiao and Sun and Dou},
+  booktitle={AAAI Conference on Artificial Intelligence},
+  year={2026}
+}
+@article{li2022geofno,
+  title={Fourier Neural Operator with Learned Deformations for PDEs on General Geometries},
+  author={Li, Zongyi and others},
+  journal={arXiv:2207.05209},
+  year={2022}
+}
+```
+
+## Model Card Authors
+
+Efradeca.
+
+## Model Card Contact
+
+https://huggingface.co/Efradeca