diff --git a/ThermalSolver/model/finite_difference.py b/ThermalSolver/model/finite_difference.py
index 05af1c9..fe50eb3 100644
--- a/ThermalSolver/model/finite_difference.py
+++ b/ThermalSolver/model/finite_difference.py
@@ -1,7 +1,6 @@
 import torch
 import torch.nn as nn
 from torch_geometric.nn import MessagePassing
-from tqdm import tqdm
 
 
 class FiniteDifferenceStep(MessagePassing):
@@ -14,8 +13,9 @@ class FiniteDifferenceStep(MessagePassing):
         assert (
             aggr == "add"
         ), "Per somme pesate, l'aggregazione deve essere 'add'."
-        self.root_weight = float(root_weight)
-        self.p = torch.nn.Parameter(torch.tensor(0.5))
+        # self.root_weight = float(root_weight)
+        self.p = torch.nn.Parameter(torch.tensor(0.8))
+        self.a = root_weight
 
     def forward(self, x, edge_index, edge_attr, deg):
         """
@@ -43,7 +43,9 @@ class FiniteDifferenceStep(MessagePassing):
         """
         TODO: add docstring.
         """
-        return self.root_weight * aggr_out + (1 - self.root_weight) * x
+        a = torch.clamp(self.a, 0.0, 1.0)
+        return a * aggr_out + (1 - a) * x
+        # return self.a * aggr_out + (1 - self.a) * x
 
 
 class GraphFiniteDifference(nn.Module):
diff --git a/ThermalSolver/model/learnable_finite_difference.py b/ThermalSolver/model/learnable_finite_difference.py
index e4f44f1..8dc6b54 100644
--- a/ThermalSolver/model/learnable_finite_difference.py
+++ b/ThermalSolver/model/learnable_finite_difference.py
@@ -1,7 +1,7 @@
 import torch
 import torch.nn as nn
 from torch_geometric.nn import MessagePassing
-from tqdm import tqdm
+from torch.nn.utils import spectral_norm
 
 
 class FiniteDifferenceStep(MessagePassing):
@@ -14,14 +14,29 @@ class FiniteDifferenceStep(MessagePassing):
         assert (
             aggr == "add"
         ), "Per somme pesate, l'aggregazione deve essere 'add'."
-        self.root_weight = float(root_weight)
+
         self.correction_net = nn.Sequential(
-            nn.Linear(2, 16),
-            nn.GELU(),
-            nn.Linear(16, 1),
+            nn.Linear(2, 6),
+            nn.Tanh(),
+            nn.Linear(6, 1),
+            nn.Tanh(),
+        )
+        self.update_net = nn.Sequential(
+            spectral_norm(nn.Linear(1, 6)),
+            nn.Softplus(),
+            spectral_norm(nn.Linear(6, 1)),
             nn.Softplus(),
         )
 
+        self.message_net = nn.Sequential(
+            spectral_norm(nn.Linear(1, 6)),
+            nn.Softplus(),
+            spectral_norm(nn.Linear(6, 1)),
+            nn.Softplus(),
+        )
+        self.p = torch.nn.Parameter(torch.tensor(0.5))
+        # self.a = torch.nn.Parameter(torch.tensor(root_weight))
+
     def forward(self, x, edge_index, edge_attr, deg):
         """
         TODO: add docstring.
@@ -33,9 +48,11 @@ class FiniteDifferenceStep(MessagePassing):
         """
         TODO: add docstring.
         """
-        x_in = torch.cat([x_j, edge_attr.view(-1, 1)], dim=-1)
-        correction = self.correction_net(x_in)
-        return edge_attr.view(-1, 1) * x_j + correction
+        # x_in = torch.cat([x_j, edge_attr.view(-1, 1)], dim=-1)
+        # correction = self.correction_net(x_in)
+        # p = torch.sigmoid(self.p)
+        # return (p * edge_attr.view(-1, 1) + (1 - p) * correction) * x_j
+        return edge_attr.view(-1, 1) * x_j
 
     def aggregate(self, inputs, index, deg):
         """
@@ -49,7 +66,7 @@ class FiniteDifferenceStep(MessagePassing):
         """
         TODO: add docstring.
         """
-        return self.root_weight * aggr_out + (1 - self.root_weight) * x
+        return self.update_net(aggr_out)
 
 
 class GraphFiniteDifference(nn.Module):
@@ -99,27 +116,14 @@ class GraphFiniteDifference(nn.Module):
         c_ij = self._compute_c_ij(c, edge_index)
         edge_attr = edge_attr * c_ij
         deg = self._compute_deg(edge_index, edge_attr, x.size(0))
-
-        # Calcola la soglia staccando x dal grafo
         conv_thres = self.threshold * torch.norm(x.detach())
 
         for _i in range(self.max_iters):
             out = self.fd_step(x, edge_index, edge_attr, deg)
             out[boundary_mask] = boundary_values.unsqueeze(-1)
-
-            # Controllo convergenza senza tracciamento gradienti
             with torch.no_grad():
                 residual_norm = torch.norm(out - x)
-
             if residual_norm < conv_thres:
                 break
-
-            # --- OTTIMIZZAZIONE CHIAVE ---
-            # Stacca 'out' dal grafo prima della prossima iterazione
-            # per evitare BPTT e risparmiare memoria.
             x = out.detach()
-
-        # Il 'out' finale restituito mantiene i gradienti
-        # dell'ULTIMA chiamata a fd_step, permettendo al modello
-        # di apprendere correttamente.
         return out, _i + 1