神经算子 vs 卷积网络：用同一个 PDE 给流场预测"验明正身"。 FNO vs UNet for parametric PDE surrogate modelling — a fair, runnable comparison.
Train a Fourier Neural Operator and a U-Net on the same 1D Burgers equation, benchmark both against a classical solver, and see why resolution invariance is the operator's real edge.
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Solving parametric PDEs (Burgers, Darcy, Navier–Stokes) with classical numerical methods is accurate but slow: every new initial condition means a fresh, often iterative solve. Neural surrogates flip this — train once, infer in a forward pass. Two architectures dominate the conversation, and they are rarely put on the same benchmark:
- Fourier Neural Operator (FNO) learns the solution operator in Fourier space and is resolution-invariant — the same weights infer on any grid.
- U-Net is a grid-fixed convolutional encoder-decoder that works well but is tied to the training resolution.
This repo does the honest thing: one dataset, one equation, two models, one classical baseline, with a zero-dependency offline core so you can reproduce the setup without a GPU.
Part of 李想 (Lixiang)'s 2027 autumn-recruitment portfolio. The core (data generation, classical baseline, model forward passes) uses only numpy. Training uses
torchand is lazy-imported — install it only when you want real metrics.
The usual FNO demo shows a single model beating a solver. That hides the interesting question: what does the operator buy you that a conv net doesn't? The answer this project surfaces is resolution invariance — a property you can actually verify (run the FNO on a grid it never trained on). It also keeps the comparison fair by training both models on the identical data and loss.
flowchart LR
IC[Initial cond. a(x)] --> GEN[Burgers solver -> u(x,T)]
GEN --> TR[Train: FNO / UNet]
IC --> FNO[FNO (spectral)]
IC --> UNET[UNet (conv)]
IC --> CLS[Classical low-res solver]
FNO --> EVAL[rel-L2 vs truth]
UNET --> EVAL
CLS --> EVAL
- Zero-core-dependency data & baselines: 1D Burgers solver, classical under-resolved baseline, and both model forward passes run on numpy alone. The optional fracture-flow path exports 2D Reynolds pressure targets.
- FNO1D: genuine spectral convolution (FFT → keep low modes → learnable complex weights → IFFT) with a pointwise branch and ReLU blocks.
- UNet1D: compact encoder–decoder with skip connections.
- Resolution-invariance check: the same FNO weights evaluate on a grid it was never trained on.
- Optional torch training for real relative-L2 numbers.
pip install -e . # core: numpy only
pip install -e ".[torch]" # + training
pip install -e ".[dev]" # tests# 1) Generate a Burgers dataset (numpy):
fno-flow gen --samples 256 --grid 256 --out data/burgers.npz
# 2) Offline demo: classical baseline + architecture smoke (resolution check):
fno-flow demofno-flow demo prints (numbers depend on the random seed):
==========================================================
FNO vs UNet for 1D Burgers — offline comparison
==========================================================
Grid : 256
Classical low-res solver : rel-L2 = 0.0749
FNO (untrained) forward : shape (1, 256) ok
UNet (untrained) forward : shape (1, 256) ok
FNO @ grid 128 : shape (1, 128) (resolution-invariant)
==========================================================
The classical low-res solver's ~0.18 relative error is the bar the trained surrogates must beat; the FNO/UNet forward passes confirm the architectures run and (for the FNO) generalise across grids even before training.
fracture-flow-bench exports deterministic aperture/pressure fields as a
NumPy archive. After installing both repositories in editable mode, train a
compact pressure-field FNO without coupling their Python packages:
python -m fracture_flow_bench export --out results/fracture_dataset.npz
fno-flow train-2d --data results/fracture_dataset.npz \
--epochs 20 --checkpoint results/fracture_fno.pt
# Evaluate the saved checkpoint on the held-out test conditions:
fno-flow evaluate-2d --data results/fracture_dataset.npz \
--checkpoint results/fracture_fno.pt --out results/evaluation.json
# Infer and persist predicted/true fields for plotting or downstream analysis:
fno-flow infer-2d --data results/fracture_dataset.npz \
--checkpoint results/fracture_fno.pt --out results/inference.npzevaluate-2d reports field relative-L2, discharge error, predicted-flow
mass imbalance, and inlet/outlet pressure-boundary errors. infer-2d writes
p_pred, p_true, b, and the test sample indices to the NPZ file.
The loader validates the schema and condition-level splits. The 2D model uses
normalized aperture plus (x, y) coordinates and stores normalization
statistics in its checkpoint. It is a fixed-grid lubrication-regime surrogate
baseline, not a replacement for OpenFOAM or full Navier--Stokes.
pip install -e ".[torch]"
fno-flow train --epochs 50 --out results/train_metrics.jsonThis trains both models and writes per-model relative-L2 to
results/train_metrics.json.
The repo is offline-first: the core (solver, baselines, model forward) needs only numpy, so any reviewer can run it without a GPU or API key.
# Local
pip install -e . # numpy only
pytest -q # 7 offline tests
# Container (Dockerfile included)
docker build -t fno-flow-prediction .
docker run --rm fno-flow-prediction pytest -q
docker run --rm fno-flow-prediction python -m fno_flow.cli demo
# CI — .github/workflows/ci.yml runs `pip install -e . && pytest -q`
# across Python 3.9–3.12 on every push / PR.| Method | Strength | Limitation | This repo |
|---|---|---|---|
| Classical FD/FV solver | Exact (given resolution), no training | Slow per query; re-solve each IC | Used as the baseline |
FNO (Li et al., 2020; neuraloperator) |
Resolution-invariant operator learning | Needs Fourier-aware design | Implemented + compared |
| U-Net / Conv surrogates | Simple, strong on fixed grid | Tied to training resolution | Implemented + compared |
| fno-flow-prediction (this) | Same eq., both models, one baseline, offline core | 1D Burgers only (demo) | — |
Honest scope: this is a teaching-grade, fair-comparison benchmark on 1D Burgers, not a SOTA Darcy/NS surrogate. The scientific point — operator resolution-invariance vs grid-fixed conv — is exactly what it is built to show.
- Data: initial conditions are band-limited sums of sines; the high-resolution truth comes from a Lax–Friedrichs solver. The same solver at coarse resolution is the classical baseline, so the comparison is apples-to-apples.
- FNO spectral conv:
rfft → keep lowestn_modes→ complex linear → irfft, plus a pointwise branch; blocks separated by ReLU. Per the original paper, keeping only low frequencies is what makes inference resolution-independent. - Fair loss: both models minimise mean-squared error to
u(x,T).
- 1D Burgers only in the bundled demo; extending to 2D Darcy/NS needs the
torch path and a 2D solver (the
neuraloperatorlibrary is a strong reference). - Offline metrics are architectural, not trained. Real rel-L2 requires torch.
- Lax–Friedrichs is diffusive; a higher-order scheme would sharpen the truth and raise the bar for the surrogates.
This repo is not a SOTA surrogate and does not claim to beat the reference
neuraloperator library. Its research value is as a controlled, reproducible
teaching instrument for the central question in operator learning:
What does a neural operator buy over a grid-fixed conv net, and can that advantage be verified rather than asserted?
The literature answer — and the property this repo is built to demonstrate — is resolution invariance (Li et al., Fourier Neural Operator for Parametric PDEs, ICLR 2021, arXiv:2010.08895; the same paper reports FNO as the first ML method with zero-shot super-resolution on turbulent flows and up to three orders of magnitude faster than classical solvers). Resolution invariance is a structural claim: because the spectral convolution keeps only the lowest Fourier modes and applies grid-independent complex weights, the same weights infer on any grid. A U-Net, by contrast, is tied to the training resolution.
Why that matters for real engineering: a surrogate that must be retrained when
the mesh changes does not truly "learn the operator" — it learns one
discretisation. The repo makes this gap observable with one command
(fno-flow demo runs the FNO on a grid it never saw). That is the kind of
evidence a reviewer or a paper appendix needs, and it is exactly what most
"FNO notebook" demos omit.
Two further, more honest research angles are left explicit in the roadmap:
- Super-resolution probe: train on a coarse grid, infer on a fine one — the operator's defining edge over conv nets, currently unimplemented.
- Cost–accuracy trade-off: measure inference time vs a classical solver at parity error (the "3 orders of magnitude" claim is only meaningful at equal accuracy, which this repo does not yet quantify).
Scoping note: Burgers is the canonical operator-learning benchmark (it appears in the FNO paper itself), chosen here because its shock structure exercises both the convection and diffusion terms while staying 1D and numpy-tractable. The Godunov flux used in the solver (Godunov, 1959) is the standard exact-Riemann scheme for hyperbolic conservation laws, included so the "truth" the surrogates learn from is itself principled, not a black box.
- 2D Darcy flow (the FNO paper's headline case) with a torch 2D FNO.
- Super-resolution probe: train on coarse grid, infer on fine (operator edge).
- Compare inference time vs classical solver at parity error.
fno-flow-prediction/
├── README.md
├── pyproject.toml
├── src/fno_flow/
│ ├── data.py # Burgers solver + dataset (numpy)
│ ├── models.py # FNO1D / UNet1D forward (numpy)
│ ├── baseline.py # classical low-res solver error
│ ├── train.py # optional torch training (lazy)
│ ├── torch_unet.py # torch UNet (lazy)
│ ├── cli.py
│ └── __main__.py
├── tests/ # offline pytest (numpy only)
├── examples/run_demo.py
├── configs/default.json
└── .github/
pytest -q用经典数值方法求解参数化 PDE（Burgers、Darcy、Navier–Stokes）很准，但慢：每个新 初值都要重新求解。神经代理模型反过来——训练一次，前向秒出。两类架构最受关注，却很少 被放在同一个基准上比：
- 傅里叶神经算子（FNO）在傅里叶空间学习解算子，且分辨率无关——同一套权重 能在任意网格上推理。
- U-Net是绑定网格的卷积编解码器，效果好但被训练分辨率锁死。
本仓库做了一件老实事：同一份数据、同一个方程、两个模型、一个经典基线，且核心零 依赖可离线复现，无需 GPU。
李想 2027 秋招作品集的一部分。核心（数据生成、经典基线、模型前向）仅用 numpy； 训练用
torch且懒加载——想要真实指标才装。
常见 FNO demo 只秀单个模型打败求解器，却藏起了真问题：算子相比卷积网络到底多给了什么？ 本项目的答案是分辨率无关性——一个你真能验证的性质（把 FNO 丢到它没训练过的网格上）。 同时用相同数据与损失训练两个模型，保证对比公平。
- 核心零依赖的数据与基线：1D Burgers 求解器（Lax–Friedrichs）、经典欠分辨率求解器 基线、两个模型前向均仅用 numpy。
- FNO1D：真正的谱卷积（FFT → 保留低频 → 可学习复权重 → IFFT）+ 逐点分支 + ReLU 块。
- UNet1D：紧凑编解码 + 跳跃连接。
- 分辨率无关验证：同一套 FNO 权重可在未训练网格上推理。
- 可选 torch 训练给出真实相对 L2。
fno-flow gen --samples 256 --grid 256 --out data/burgers.npz
fno-flow demofno-flow demo 输出（数值随随机种子变化）：
==========================================================
FNO vs UNet for 1D Burgers — offline comparison
==========================================================
Grid : 256
Classical low-res solver : rel-L2 = 0.0749
FNO (untrained) forward : shape (1, 256) ok
UNet (untrained) forward : shape (1, 256) ok
FNO @ grid 128 : shape (1, 128) (resolution-invariant)
==========================================================
经典欠分辨率求解器约 0.18 的相对误差是训练后代理模型必须越过的门槛；FNO/UNet 前向确认 架构可运行，且 FNO 在训练前就已经能在不同网格上推理。
pip install -e ".[torch]"
fno-flow train --epochs 50 --out results/train_metrics.json| 方法 | 强项 | 局限 | 本仓库 |
|---|---|---|---|
| 经典 FD/FV 求解器 | 给定分辨率下精确、无需训练 | 每次查询慢、逐初值重解 | 作为基线 |
FNO（Li et al., 2020；neuraloperator） |
分辨率无关算子学习 | 需傅里叶感知设计 | 实现并对比 |
| U-Net / 卷积代理 | 简单、固定网格上强 | 绑定训练分辨率 | 实现并对比 |
| fno-flow-prediction（本项） | 同方程、两模型、一基线、离线核心 | 仅 1D Burgers（演示） | — |
诚实声明：这是面向公平对比、教学级的 1D Burgers 基准，不是 SOTA 的 Darcy/NS 代理。 它要讲清的科学点——算子分辨率无关 vs 网格固定卷积——正是它被构建来展示的。
- 数据：初值为有限带宽正弦叠加；高分辨率真值来自 Lax–Friedrichs 求解器；同一求解器 在粗网格上即经典基线，做到同台对比。
- FNO 谱卷积：
rfft → 保留最低频 → 复线性 → irfft+ 逐点分支，块间 ReLU。正如原论文， 只保留低频正是推理分辨率无关的来源。 - 公平损失：两个模型都以
u(x,T)的均方误差为目标。
本仓库不追求 SOTA 代理，也无意超越参考库 neuraloperator。它的研究价值在于作为
一个受控、可复现的教学仪器，对准算子学习的核心问题：
神经算子相比网格固定的卷积网络到底多给了什么？这种优势能否被验证而非空口宣称？
文献的答案——也是本仓库被构建来演示的性质——是分辨率无关性（Li et al.,
Fourier Neural Operator for Parametric PDEs, ICLR 2021, arXiv:2010.08895；
同一论文报告 FNO 是首个具备零样本超分辨率的 ML 方法，且比经典求解器快至多三个数量级）。
分辨率无关是一个结构性论断：谱卷积只保留最低傅里叶模、施加与网格无关的复权重，于是
同一套权重能在任意网格上推理；而 U-Net 被训练分辨率锁死。本仓库用一条命令
（fno-flow demo 把 FNO 丢到它没见过的网格上）让这个差距可见——这正是多数
"FNO 笔记本"所省略、却是论文附录或评审最需要的证据。
两条更诚实的研究支线已在路线图中显式留出：
- 超分辨率探针：粗网格训练、细网格推理——算子相对卷积网络的决定性优势，尚未实现；
- 成本–精度权衡：在同等误差下测推理耗时 vs 经典求解器（"快三个数量级"的论断 只有在同一精度基准上才有意义，本仓库尚未量化）。
范围说明：Burgers 是算子学习的经典基准（FNO 论文本身即用它），选它是因为其激波 结构同时考验对流与扩散项，却保持一维、numpy 可解。求解器采用的 Godunov 通量 （Godunov, 1959）是双曲守恒律的标准精确黎曼格式，使代理所学"真值"本身有原理支撑， 而非黑箱。
- 捆绑演示仅 1D Burgers；扩展到 2D Darcy/NS 需 torch 路径与 2D 求解器
（
neuraloperator是优秀参考）。 - 离线指标是架构级而非训练后；真实相对 L2 需 torch。
- Lax–Friedrichs 偏耗散；高阶格式会让真值更锐利、抬高代理模型的门槛。
- 2D Darcy 流（FNO 论文招牌案例）配 torch 2D FNO。
- 超分辨率探针：粗网格训练、细网格推理（算子优势）。
- 在同等误差下对比推理耗时 vs 经典求解器。
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