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#!/usr/bin/env python
# coding: utf-8
# # Tutorial: Reduced Order Model with Graph Neural Networks
# [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/mathLab/PINA/blob/master/tutorials/tutorial22/tutorial.ipynb)
# > ##### ⚠️ ***Before starting:***
# > 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.
# 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).
# Let's start by importing the useful modules:
## routine needed to run the notebook on Google Colab
    import google.colab
    IN_COLAB = True
    IN_COLAB = False
if IN_COLAB:
    get_ipython().system('pip install "pina-mathlab[tutorial]"')
    get_ipython().system('wget "https://github.com/mathLab/PINA/raw/refs/heads/master/tutorials/tutorial22/holed_poisson.pt" -O "holed_poisson.pt"')
import torch
from torch import nn
from torch_geometric.nn import GMMConv
from torch_geometric.data import (
)  # alternatively, from pina.graph import Graph, LabelBatch
from torch_geometric.utils import to_dense_batch
import matplotlib.pyplot as plt
import warnings
warnings.filterwarnings("ignore")
from pina import Trainer
from pina.model import FeedForward
from pina.optim import TorchOptimizer
from pina.solver import ReducedOrderModelSolver
from pina.problem.zoo import SupervisedProblem
# ## Data Generation
# In this tutorial, we will focus on solving the parametric **Poisson** equation, a linear PDE. The equation is given by:
# \begin{cases}
# -\frac{1}{10}\Delta u = 1, &\Omega(\boldsymbol{\mu}),\\
# u = 0, &\partial \Omega(\boldsymbol{\mu}).
# \end{cases}
# 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 $\boldsymbol{\mu}$. 
# 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. 
# 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.
# **Note:**
# The numerical integration is obtained using a finite element method with the [RBniCS library](https://www.rbnicsproject.org/).
# === load the data ===
# x, y -> spatial discretization
# edge_index, triang -> connectivity matrix, triangulation
# u, params -> solution field, parameters
data = torch.load("holed_poisson.pt")
x = data["x"]
y = data["y"]
edge_index = data["edge_index"]
u = data["u"]
triang = data["triang"]
params = data["mu"]
# simple plot
plt.figure(figsize=(4, 4))
plt.tricontourf(x[:, 10], y[:, 10], triang, u[:, 10], 100, cmap="jet")
plt.scatter(params[10, 0], params[10, 1], c="r", marker="x", s=100)
plt.tight_layout()
# ## Graph-Based Reduced Order Modeling
# 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.
# <p align="center">
#     <img src="http://raw.githubusercontent.com/mathLab/PINA/master/tutorials/static/gca_off_on_3_pina.png" alt="GCA-ROM" width="800"/>
# 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.
# Formally, the autoencoder consists of an **encoder** $\mathcal{E}$, a **decoder** $\mathcal{D}$, and a **parametric mapping** $\mathcal{M}$:
# z = \mathcal{E}(u(\mathbf{x}, \boldsymbol{\mu})), 
# \hat{u}(\mathbf{x}, \boldsymbol{\mu}) = \mathcal{D}(z),
# \hat{z} = \mathcal{M}(\boldsymbol{\mu}),
# 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*.
# The training objective balances two terms:
# 1. **Reconstruction loss**: ensuring the autoencoder can faithfully reconstruct $u$ from $z$.
# 2. **Latent consistency loss**: enforcing that the parametric map $\mathcal{M}(\boldsymbol{\mu})$ approximates the encoder’s latent space.
# The combined loss function is:
# \mathcal{L}(\theta) = \frac{1}{N} \sum_{i=1}^N 
# \big\| u(\mathbf{x}, \boldsymbol{\mu}_i) - 
# \mathcal{D}\!\big(\mathcal{E}(u(\mathbf{x}, \boldsymbol{\mu}_i))\big) 
# \big\|_2^2
# \;+\; \frac{1}{N} \sum_{i=1}^N
# \big\| \mathcal{E}(u(\mathbf{x}, \boldsymbol{\mu}_i)) - \mathcal{M}(\boldsymbol{\mu}_i) \big\|_2^2.
# This framework leverages the expressive power of GNNs for unstructured geometries and the efficiency of ROMs for handling parametric PDEs.
# We will now build the autoencoder network, which is a `nn.Module` with two methods: `encode` and `decode`.
class GraphConvolutionalAutoencoder(nn.Module):
    def __init__(
        self, hidden_channels, bottleneck, input_size, ffn, act=nn.ELU
        super().__init__()
        self.hidden_channels, self.input_size = hidden_channels, input_size
        self.act = act()
        self.current_graph = None
        # Encoder GMM layers
        self.fc_enc1 = nn.Linear(input_size * hidden_channels[-1], ffn)
        self.fc_enc2 = nn.Linear(ffn, bottleneck)
        self.encoder_convs = nn.ModuleList(
                GMMConv(
                    hidden_channels[i],
                    hidden_channels[i + 1],
                    kernel_size=5,
                for i in range(len(hidden_channels) - 1)
        # Decoder GMM layers
        self.fc_dec1 = nn.Linear(bottleneck, ffn)
        self.fc_dec2 = nn.Linear(ffn, input_size * hidden_channels[-1])
        self.decoder_convs = nn.ModuleList(
                GMMConv(
                    hidden_channels[-i - 1],
                    hidden_channels[-i - 2],
                    kernel_size=5,
                for i in range(len(hidden_channels) - 1)
    def encode(self, data):
        self.current_graph = data
        x = data.x
        h = x
        for conv in self.encoder_convs:
            x = self.act(conv(x, data.edge_index, data.edge_weight) + h)
        x = x.reshape(
            data.num_graphs, self.input_size * self.hidden_channels[-1]
        return self.fc_enc2(self.act(self.fc_enc1(x)))
    def decode(self, z, decoding_graph=None):
        data = decoding_graph or self.current_graph
        x = self.act(self.fc_dec2(self.act(self.fc_dec1(z)))).reshape(
            data.num_graphs * self.input_size, self.hidden_channels[-1]
        h = x
        for i, conv in enumerate(self.decoder_convs):
            x = conv(x, data.edge_index, data.edge_weight) + h
            if i != len(self.decoder_convs) - 1:
                x = self.act(x)
        return x
# Great! We now need to build the graph structure (a PyTorch Geometric `Data` object) from the numerical solver outputs.
# 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:
# - **Nodes** correspond to spatial points in the mesh. Each node stores the **solution value** $u$ at that point as a feature.  
# - **Edges** represent mesh connectivity. For each edge, we compute:
#   - **Edge attributes**: the relative displacement vector between the two nodes.  
#   - **Edge weights**: the Euclidean distance between the connected nodes.  
# - **Positions** store the physical $(x, y)$ coordinates of the nodes.
# For each parameter realization $\boldsymbol{\mu}_i$, we therefore construct a PyTorch Geometric `Data` object:
# number of nodes and number of graphs (parameter realizations)
num_nodes, num_graphs = u.shape
graphs = []
for g in range(num_graphs):
    # node positions
    pos = torch.stack([x[:, g], y[:, g]], dim=1)  # shape [num_nodes, 2]
    # edge attributes and weights
    ei, ej = pos[edge_index[0]], pos[edge_index[1]]  # [num_edges, 2]
    edge_attr = torch.abs(ej - ei)  # relative offsets
    edge_weight = edge_attr.norm(p=2, dim=1, keepdim=True)  # Euclidean distance
    # node features (solution values)
    node_features = u[:, g].unsqueeze(-1)  # [num_nodes, 1]
    # build PyG graph
    graphs.append(
        Data(
            x=node_features,
            edge_index=edge_index,
            edge_weight=edge_weight,
            edge_attr=edge_attr,
            pos=pos,
# ## Training with PINA
# 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:  
# - **Input**: the parameter tensor $\boldsymbol{\mu}$ describing each scenario.  
# - **Output**: the corresponding graph structure (PyTorch Geometric `Data` object) that we aim to reconstruct.  
problem = SupervisedProblem(params, graphs)
# Next, we build the **autoencoder network** and the **interpolation network**.  
# - The **Graph Convolutional Autoencoder (GCA)** encodes the high-dimensional graph data into a compact latent space and reconstructs the graphs from this latent representation.  
# - 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.
reduction_network = GraphConvolutionalAutoencoder(
    hidden_channels=[1, 1], bottleneck=8, input_size=1352, ffn=200, act=nn.ELU
interpolation_network = FeedForward(
    input_dimensions=2,
    output_dimensions=8,
    n_layers=2,
    inner_size=200,
    func=nn.Tanh,
# 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.  
# This solver requires two components:  
# - an **interpolation network**, which maps parameters $\boldsymbol{\mu}$ to the latent space, and  
# - a **reduction network**, which in our case is the **autoencoder** that compresses and reconstructs the graph data.  
# This loss handles both Data and Torch.Tensors
class CustomMSELoss(nn.MSELoss):
    def forward(self, output, target):
        if isinstance(output, Data):
            output = output.x
        if isinstance(target, Data):
            target = target.x
        return torch.nn.functional.mse_loss(
            output, target, reduction=self.reduction
# Define the solver
solver = ReducedOrderModelSolver(
    problem=problem,
    reduction_network=reduction_network,
    interpolation_network=interpolation_network,
    use_lt=False,
    loss=CustomMSELoss(),
    optimizer=TorchOptimizer(torch.optim.Adam, lr=0.001, weight_decay=1e-05),
# 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.
trainer = Trainer(
    solver=solver,
    accelerator="cpu",
    max_epochs=300,
    train_size=0.3,
    val_size=0.7,
    test_size=0.0,
    shuffle=True,
trainer.train()
# Once the model is trained, we can test the reconstruction by following two steps:
# 1. **Interpolate**: Use the `interpolation_network` to map a new parameter $\boldsymbol{\mu}^*$ to the latent space.  
# 2. **Decode**: Pass the interpolated latent vector through the autoencoder (`reduction_network`) to reconstruct the corresponding graph data.
# interpolate
z = interpolation_network(params)
batch = Batch.from_data_list(graphs)
out = reduction_network.decode(z, decoding_graph=batch)
out, _ = to_dense_batch(out, batch.batch)
out = out.squeeze(-1).T.detach()
# Let's compute the total error, and plot a sample solution:
# compute error
l2_error = (torch.norm(out - u, dim=0) / torch.norm(u, dim=0)).mean()
print(f"L2 relative error {l2_error:.2%}")
# plot solution
idx_to_plot = 42
# Determine min and max values for color scaling
vmin = min(out[:, idx_to_plot].min(), u[:, idx_to_plot].min())
vmax = max(out[:, idx_to_plot].max(), u[:, idx_to_plot].max())
plt.figure(figsize=(16, 4))
plt.subplot(1, 3, 1)
plt.tricontourf(
    x[:, idx_to_plot],
    y[:, idx_to_plot],
    triang,
    out[:, idx_to_plot],
    cmap="jet",
    vmin=vmin,
    vmax=vmax,
plt.title("GCA-ROM")
plt.colorbar()
plt.subplot(1, 3, 2)
plt.title("True")
plt.tricontourf(
    x[:, idx_to_plot],
    y[:, idx_to_plot],
    triang,
    u[:, idx_to_plot],
    cmap="jet",
    vmin=vmin,
    vmax=vmax,
plt.colorbar()
plt.subplot(1, 3, 3)
plt.title("Square Error")
plt.tricontourf(
    x[:, idx_to_plot],
    y[:, idx_to_plot],
    triang,
    (u - out).pow(2)[:, idx_to_plot],
    cmap="jet",
plt.colorbar()
plt.ticklabel_format()
# Nice! We can see that the network is correctly learning the solution operator, and the workflow was very straightforward.  
# 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.
# ## What's Next?
# 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:
# 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.
# 2. **Explore Physical Constraints** — Incorporate physics-informed terms or constraints during training to improve model generalization and ensure physically consistent predictions.
# 3. **...and many more!** — The possibilities are vast! Continue experimenting with advanced configurations, solvers, and features in PINA.
# For more resources and tutorials, check out the [PINA Documentation](https://mathlab.github.io/PINA/).
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