Co-authored-by: dario-coscia <dario-coscia@users.noreply.github.com>
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tutorials/tutorial22/tutorial.py
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tutorials/tutorial22/tutorial.py
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#!/usr/bin/env python
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# coding: utf-8
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# # Tutorial: Reduced Order Model with Graph Neural Networks
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#
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# [](https://colab.research.google.com/github/mathLab/PINA/blob/master/tutorials/tutorial22/tutorial.ipynb)
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#
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#
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# > ##### ⚠️ ***Before starting:***
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# > We assume you are already familiar with the concepts covered in the [Data Structure for SciML](https://mathlab.github.io/PINA/tutorial19/tutorial.html) tutorial. If not, we strongly recommend reviewing them before exploring this advanced topic.
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#
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# In this tutorial, we will demonstrate a typical use case of **PINA** for Reduced Order Modelling using Graph Convolutional Neural Network. The tutorial is largely inspired by the paper [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://www.sciencedirect.com/science/article/pii/S0021999124000111).
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#
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# Let's start by importing the useful modules:
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# In[ ]:
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## routine needed to run the notebook on Google Colab
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try:
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import google.colab
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IN_COLAB = True
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except:
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IN_COLAB = False
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if IN_COLAB:
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get_ipython().system('pip install "pina-mathlab[tutorial]"')
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get_ipython().system('wget "https://github.com/mathLab/PINA/raw/refs/heads/master/tutorials/tutorial22/holed_poisson.pt" -O "holed_poisson.pt"')
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import torch
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from torch import nn
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from torch_geometric.nn import GMMConv
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from torch_geometric.data import (
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Data,
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Batch,
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) # alternatively, from pina.graph import Graph, LabelBatch
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from torch_geometric.utils import to_dense_batch
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import matplotlib.pyplot as plt
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import warnings
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warnings.filterwarnings("ignore")
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from pina import Trainer
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from pina.model import FeedForward
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from pina.optim import TorchOptimizer
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from pina.solver import ReducedOrderModelSolver
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from pina.problem.zoo import SupervisedProblem
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# ## Data Generation
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#
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# In this tutorial, we will focus on solving the parametric **Poisson** equation, a linear PDE. The equation is given by:
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#
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# $$
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# \begin{cases}
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# -\frac{1}{10}\Delta u = 1, &\Omega(\boldsymbol{\mu}),\\
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# u = 0, &\partial \Omega(\boldsymbol{\mu}).
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# \end{cases}
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# $$
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#
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# In this equation, $\Omega(\boldsymbol{\mu}) = [0, 1]\times[0,1] \setminus [\mu_1, \mu_2]\times[\mu_1+0.3, \mu_2+0.3]$ represents the spatial domain characterized by a parametrized hole defined via $\boldsymbol{\mu} = (\mu_1, \mu_2) \in \mathbb{P} = [0.1, 0.6]\times[0.1, 0.6]$. Thus, the geometrical parameters define the left bottom corner of a square obstacle of dimension $0.3$. The problem is coupled with homogenous Dirichlet conditions on both internal and external boundaries. In this setting, $u(\mathbf{x}, \boldsymbol{\mu})\in \mathbb{R}$ is the value of the function $u$ at each point in space for a specific parameter $\boldsymbol{\mu}$.
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#
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# We have already generated data for different parameters. The dataset is obtained via $\mathbb{P}^1$ FE method, and an equispaced sampling with 11 points in each direction of the parametric space.
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#
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# The goal is to build a Reduced Order Model that given a new parameter $\boldsymbol{\mu}^*$, is able to get the solution $u$ *for any discretization* $\mathbf{x}$. To this end, we will train a Graph Convolutional Autoencoder Reduced Order Model (GCA-ROM), as presented in [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://www.sciencedirect.com/science/article/pii/S0021999124000111). We will cover the architecture details later, but for now, let’s start by importing the data.
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#
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# **Note:**
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# The numerical integration is obtained using a finite element method with the [RBniCS library](https://www.rbnicsproject.org/).
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# In[21]:
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# === load the data ===
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# x, y -> spatial discretization
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# edge_index, triang -> connectivity matrix, triangulation
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# u, params -> solution field, parameters
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data = torch.load("holed_poisson.pt")
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x = data["x"]
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y = data["y"]
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edge_index = data["edge_index"]
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u = data["u"]
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triang = data["triang"]
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params = data["mu"]
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# simple plot
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plt.figure(figsize=(4, 4))
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plt.tricontourf(x[:, 10], y[:, 10], triang, u[:, 10], 100, cmap="jet")
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plt.scatter(params[10, 0], params[10, 1], c="r", marker="x", s=100)
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plt.tight_layout()
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plt.show()
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# ## Graph-Based Reduced Order Modeling
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#
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# In this problem, the geometry of the spatial domain is **unstructured**, meaning that classical grid-based methods (e.g., CNNs) are not well suited. Instead, we represent the mesh as a **graph**, where nodes correspond to spatial degrees of freedom and edges represent connectivity. This makes **Graph Neural Networks (GNNs)**, and in particular **Graph Convolutional Networks (GCNs)**, a natural choice to process the data.
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#
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# <p align="center">
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# <img src="http://raw.githubusercontent.com/mathLab/PINA/master/tutorials/static/gca_off_on_3_pina.png" alt="GCA-ROM" width="800"/>
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# </p>
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#
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# To reduce computational complexity while preserving accuracy, we employ a **Reduced Order Modeling (ROM)** strategy (see picture above). The idea is to map high-dimensional simulation data $u(\mathbf{x}, \boldsymbol{\mu})$ to a compact **latent space** using a **graph convolutional encoder**, and then reconstruct it back via a **decoder** (offline phase). The latent representation captures the essential features of the solution manifold. Moreover, we can learn a **parametric map** $\mathcal{M}$ from the parameter space $\boldsymbol{\mu}$ directly into the latent space, enabling predictions for new unseen parameters.
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#
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# Formally, the autoencoder consists of an **encoder** $\mathcal{E}$, a **decoder** $\mathcal{D}$, and a **parametric mapping** $\mathcal{M}$:
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# $$
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# z = \mathcal{E}(u(\mathbf{x}, \boldsymbol{\mu})),
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# \quad
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# \hat{u}(\mathbf{x}, \boldsymbol{\mu}) = \mathcal{D}(z),
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# \quad
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# \hat{z} = \mathcal{M}(\boldsymbol{\mu}),
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# $$
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# where $z \in \mathbb{R}^r$ is the latent representation with $r \ll N$ (the number of degrees of freedom) and the **hat notation** ($\hat{u}, \hat{z}$) indicates *learned or approximated quantities*.
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#
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# The training objective balances two terms:
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# 1. **Reconstruction loss**: ensuring the autoencoder can faithfully reconstruct $u$ from $z$.
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# 2. **Latent consistency loss**: enforcing that the parametric map $\mathcal{M}(\boldsymbol{\mu})$ approximates the encoder’s latent space.
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#
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# The combined loss function is:
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# $$
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# \mathcal{L}(\theta) = \frac{1}{N} \sum_{i=1}^N
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# \big\| u(\mathbf{x}, \boldsymbol{\mu}_i) -
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# \mathcal{D}\!\big(\mathcal{E}(u(\mathbf{x}, \boldsymbol{\mu}_i))\big)
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# \big\|_2^2
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# \;+\; \frac{1}{N} \sum_{i=1}^N
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# \big\| \mathcal{E}(u(\mathbf{x}, \boldsymbol{\mu}_i)) - \mathcal{M}(\boldsymbol{\mu}_i) \big\|_2^2.
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# $$
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# This framework leverages the expressive power of GNNs for unstructured geometries and the efficiency of ROMs for handling parametric PDEs.
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#
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# We will now build the autoencoder network, which is a `nn.Module` with two methods: `encode` and `decode`.
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#
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# In[3]:
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class GraphConvolutionalAutoencoder(nn.Module):
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def __init__(
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self, hidden_channels, bottleneck, input_size, ffn, act=nn.ELU
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):
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super().__init__()
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self.hidden_channels, self.input_size = hidden_channels, input_size
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self.act = act()
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self.current_graph = None
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# Encoder GMM layers
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self.fc_enc1 = nn.Linear(input_size * hidden_channels[-1], ffn)
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self.fc_enc2 = nn.Linear(ffn, bottleneck)
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self.encoder_convs = nn.ModuleList(
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[
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GMMConv(
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hidden_channels[i],
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hidden_channels[i + 1],
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dim=1,
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kernel_size=5,
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)
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for i in range(len(hidden_channels) - 1)
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]
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)
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# Decoder GMM layers
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self.fc_dec1 = nn.Linear(bottleneck, ffn)
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self.fc_dec2 = nn.Linear(ffn, input_size * hidden_channels[-1])
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self.decoder_convs = nn.ModuleList(
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[
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GMMConv(
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hidden_channels[-i - 1],
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hidden_channels[-i - 2],
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dim=1,
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kernel_size=5,
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)
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for i in range(len(hidden_channels) - 1)
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]
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)
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def encode(self, data):
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self.current_graph = data
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x = data.x
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h = x
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for conv in self.encoder_convs:
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x = self.act(conv(x, data.edge_index, data.edge_weight) + h)
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x = x.reshape(
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data.num_graphs, self.input_size * self.hidden_channels[-1]
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)
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return self.fc_enc2(self.act(self.fc_enc1(x)))
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def decode(self, z, decoding_graph=None):
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data = decoding_graph or self.current_graph
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x = self.act(self.fc_dec2(self.act(self.fc_dec1(z)))).reshape(
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data.num_graphs * self.input_size, self.hidden_channels[-1]
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)
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h = x
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for i, conv in enumerate(self.decoder_convs):
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x = conv(x, data.edge_index, data.edge_weight) + h
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if i != len(self.decoder_convs) - 1:
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x = self.act(x)
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return x
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# Great! We now need to build the graph structure (a PyTorch Geometric `Data` object) from the numerical solver outputs.
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#
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# The solver provides the solution values $u(\mathbf{x}, \boldsymbol{\mu})$ for each parameter instance $\boldsymbol{\mu}$, along with the node coordinates $(x, y)$ of the unstructured mesh. Because the geometry is not defined on a regular grid, we naturally represent the mesh as a graph:
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#
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# - **Nodes** correspond to spatial points in the mesh. Each node stores the **solution value** $u$ at that point as a feature.
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# - **Edges** represent mesh connectivity. For each edge, we compute:
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# - **Edge attributes**: the relative displacement vector between the two nodes.
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# - **Edge weights**: the Euclidean distance between the connected nodes.
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# - **Positions** store the physical $(x, y)$ coordinates of the nodes.
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#
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# For each parameter realization $\boldsymbol{\mu}_i$, we therefore construct a PyTorch Geometric `Data` object:
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#
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# In[4]:
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# number of nodes and number of graphs (parameter realizations)
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num_nodes, num_graphs = u.shape
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graphs = []
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for g in range(num_graphs):
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# node positions
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pos = torch.stack([x[:, g], y[:, g]], dim=1) # shape [num_nodes, 2]
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# edge attributes and weights
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ei, ej = pos[edge_index[0]], pos[edge_index[1]] # [num_edges, 2]
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edge_attr = torch.abs(ej - ei) # relative offsets
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edge_weight = edge_attr.norm(p=2, dim=1, keepdim=True) # Euclidean distance
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# node features (solution values)
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node_features = u[:, g].unsqueeze(-1) # [num_nodes, 1]
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# build PyG graph
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graphs.append(
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Data(
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x=node_features,
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edge_index=edge_index,
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edge_weight=edge_weight,
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edge_attr=edge_attr,
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pos=pos,
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)
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)
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# ## Training with PINA
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#
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# Everything is now ready! We can use **PINA** to train the model, following the workflow from previous tutorials. First, we need to define the problem. In this case, we will use the [`SupervisedProblem`](https://mathlab.github.io/PINA/_rst/problem/zoo/supervised_problem.html#module-pina.problem.zoo.supervised_problem), which expects:
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#
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# - **Input**: the parameter tensor $\boldsymbol{\mu}$ describing each scenario.
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# - **Output**: the corresponding graph structure (PyTorch Geometric `Data` object) that we aim to reconstruct.
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# In[5]:
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problem = SupervisedProblem(params, graphs)
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# Next, we build the **autoencoder network** and the **interpolation network**.
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#
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# - The **Graph Convolutional Autoencoder (GCA)** encodes the high-dimensional graph data into a compact latent space and reconstructs the graphs from this latent representation.
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# - The **interpolation network** (or parametric map) learns to map a new parameter $\boldsymbol{\mu}^*$ directly into the latent space, enabling the model to predict solutions for unseen parameter instances without running the full encoder.
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# In[6]:
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reduction_network = GraphConvolutionalAutoencoder(
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hidden_channels=[1, 1], bottleneck=8, input_size=1352, ffn=200, act=nn.ELU
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)
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interpolation_network = FeedForward(
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input_dimensions=2,
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output_dimensions=8,
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n_layers=2,
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inner_size=200,
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func=nn.Tanh,
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)
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# Finally, we will use the [`ReducedOrderModelSolver`](https://mathlab.github.io/PINA/_rst/solver/supervised_solver/reduced_order_model.html#pina.solver.supervised_solver.reduced_order_model.ReducedOrderModelSolver) to perform the training, as discussed earlier.
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#
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# This solver requires two components:
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# - an **interpolation network**, which maps parameters $\boldsymbol{\mu}$ to the latent space, and
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# - a **reduction network**, which in our case is the **autoencoder** that compresses and reconstructs the graph data.
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# In[7]:
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# This loss handles both Data and Torch.Tensors
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class CustomMSELoss(nn.MSELoss):
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def forward(self, output, target):
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if isinstance(output, Data):
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output = output.x
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if isinstance(target, Data):
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target = target.x
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return torch.nn.functional.mse_loss(
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output, target, reduction=self.reduction
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)
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# Define the solver
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solver = ReducedOrderModelSolver(
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problem=problem,
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reduction_network=reduction_network,
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interpolation_network=interpolation_network,
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use_lt=False,
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loss=CustomMSELoss(),
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optimizer=TorchOptimizer(torch.optim.Adam, lr=0.001, weight_decay=1e-05),
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)
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# Training is performed as usual using the **`Trainer`** API. In this tutorial, we will use only **30% of the data** for training, and only $300$ epochs of training to illustrate the workflow.
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# In[ ]:
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trainer = Trainer(
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solver=solver,
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accelerator="cpu",
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max_epochs=300,
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train_size=0.3,
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val_size=0.7,
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test_size=0.0,
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shuffle=True,
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)
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trainer.train()
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# Once the model is trained, we can test the reconstruction by following two steps:
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#
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# 1. **Interpolate**: Use the `interpolation_network` to map a new parameter $\boldsymbol{\mu}^*$ to the latent space.
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# 2. **Decode**: Pass the interpolated latent vector through the autoencoder (`reduction_network`) to reconstruct the corresponding graph data.
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# In[9]:
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# interpolate
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z = interpolation_network(params)
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# decode
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batch = Batch.from_data_list(graphs)
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out = reduction_network.decode(z, decoding_graph=batch)
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out, _ = to_dense_batch(out, batch.batch)
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out = out.squeeze(-1).T.detach()
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# Let's compute the total error, and plot a sample solution:
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# In[11]:
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# compute error
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l2_error = (torch.norm(out - u, dim=0) / torch.norm(u, dim=0)).mean()
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print(f"L2 relative error {l2_error:.2%}")
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# plot solution
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idx_to_plot = 42
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# Determine min and max values for color scaling
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vmin = min(out[:, idx_to_plot].min(), u[:, idx_to_plot].min())
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vmax = max(out[:, idx_to_plot].max(), u[:, idx_to_plot].max())
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plt.figure(figsize=(16, 4))
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plt.subplot(1, 3, 1)
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plt.tricontourf(
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x[:, idx_to_plot],
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y[:, idx_to_plot],
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triang,
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out[:, idx_to_plot],
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100,
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cmap="jet",
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vmin=vmin,
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vmax=vmax,
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)
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plt.title("GCA-ROM")
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plt.colorbar()
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plt.subplot(1, 3, 2)
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plt.title("True")
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plt.tricontourf(
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x[:, idx_to_plot],
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y[:, idx_to_plot],
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triang,
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u[:, idx_to_plot],
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100,
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cmap="jet",
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vmin=vmin,
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vmax=vmax,
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)
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plt.colorbar()
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plt.subplot(1, 3, 3)
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plt.title("Square Error")
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plt.tricontourf(
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x[:, idx_to_plot],
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y[:, idx_to_plot],
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triang,
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(u - out).pow(2)[:, idx_to_plot],
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100,
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cmap="jet",
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)
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plt.colorbar()
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plt.ticklabel_format()
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plt.show()
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# Nice! We can see that the network is correctly learning the solution operator, and the workflow was very straightforward.
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#
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# You may notice that the network outputs are not as smooth as the actual solution. Don’t worry — training for longer (e.g., ~5000 epochs) will produce a smoother, more accurate reconstruction.
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#
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# ## What's Next?
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#
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# Congratulations on completing the introductory tutorial on **Graph Convolutional Reduced Order Modeling**! Now that you have a solid foundation, here are a few directions to explore:
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#
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# 1. **Experiment with Training Duration** — Try different training durations and adjust the network architecture to optimize performance. Explore different integral kernels and observe how the results vary.
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#
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# 2. **Explore Physical Constraints** — Incorporate physics-informed terms or constraints during training to improve model generalization and ensure physically consistent predictions.
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#
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# 3. **...and many more!** — The possibilities are vast! Continue experimenting with advanced configurations, solvers, and features in PINA.
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#
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# For more resources and tutorials, check out the [PINA Documentation](https://mathlab.github.io/PINA/).
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