QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs
Authors: Vaibhav Mehandiratta, Saket Ramchandra
Organizations: Department of Mathematics, Birla Institute of Technology and Science, Pilani, K K Birla Goa Campus, Zuarinagar, Sancoale, Goa 403726, India
We propose QGPINNs, a physics-informed neural network framework developed in PyTorch for the numerical solution of nonlocal differential equations on quantum graphs. The framework is designed as a general computational implementation in which the solution on each edge of the graph is approximated by a neural network, while a unified graph-based loss function enforces the governing equations together with initial, boundary, and vertex transmission conditions. In particular, the formulation incorporates standard continuity and Kirchhoff-Neumann vertex conditions and Dirichlet boundary conditions into the learning process to couple the local edge-wise neural approximations into a global solution on the graph. The framework is developed for two representative classes of nonlinear models: multi-order fractional elliptic problems and time-fractional evolution equations on quantum graphs. To improve accuracy and training stability, QGPINNs integrates several graph-adapted learning strategies, including soft and hard constraint enforcement, dynamic loss balancing, Fourier feature embeddings, and a learnable singularity-capturing feature for weakly singular solutions arising in the considered problems. The framework also extends naturally to inverse problems, including the identification of the orders of fractional operators and physical parameters from noisy observational data. We validate the accuracy, computational efficiency, and physical consistency of the proposed framework through numerical experiments on benchmark graph structures and real-world networks, including the IEEE 14-bus system and an open-channel agricultural drainage network.
Figures & tables
Figure 1: Quantum graph with the degree of each vertex labeled. Node colors indicate vertex degree.
Figure 2: A star graph G with interior vertex v0 (filled) and pendant vertices v1,v2,v3 (shaded). At v0 , the solution must satisfy both the continuity condition (equal trace from all incident edges) and the Kirchhoff flux condition (outward derivatives sum to zero).
Figure 3: Graded mesh with N=12 points on time interval [0,T] . Higher r clusters points near t=0 , providing finer resolution of the initial singularity region.
Mesh Grading Factor
Soft Constraints
Hard Constraints
r=1
1.78×10−3
1.76×10−3
r=2
2.17×10−3
3.35×10−3
r=3
1.66×10−4
1.01×10−3
Table 1: Effect of the mesh grading factor r on the relative discrete L2 error ( 3.8 ) for soft and hard constraints on a metric star graph.
Figure 5
Figure 5: Dynamic evolution of the balancing parameter λ during training for different adaptive strategies.
Figure 6: Framework comparison between a standard PINN and a Fourier feature PINN. The Fourier embedding γ(x) modifies the input representation by introducing oscillatory features and thus, mitigating spectral bias and improving the learning of high-frequency components.
Strategy
Relative L2 Error
Fourier Disabled
1.90×10−1
Fourier Enabled
5.92×10−3
Table 3: Performance comparison of an embedded Fourier feature PINN with a standard PINN.
Figure 7: Performance of the network with Fourier embedding enabled and disabled. a) represents the edge residual loss, b) shows the node residual loss.
Mesh grading factor
Z(t) disabled
Z(t) enabled
r=1
2.1×10−3
9.71×10−4
r=2
2.00×10−3
7.65×10−4
r=3
2.38×10−3
1.31×10−3
r=4
2.78×10−3
1.44×10−3
Table 4: Effect of mesh grading and singularity-capturing feature on the relative L2 error
Figure 8: Comparison of PINN approximations with and without the singularity-capturing feature Z(t)=tξ .
Stage
Role
set_constraints()
Specify boundary condition type (soft/hard) and values per vertex
set_frac_scheme()
Select fractional discretization (L1 or L2-1 σ )
set_mesh()
Choose spatial/temporal collocation density and grading
set_architecture()
Configure sub-network width, depth, and Fourier features
set_singularity_capture()
Configure the singularity capturing feature.
compile()
Initialise the network architecture
Table 5: Important solver attributes. The framework also provides a secondary set of attributes to allow granular control over network architecture
Figure 9: System architecture and pipeline flow of QGPINNs.
Figure 11: (a) Computational demand of each variant. (b) A 3-Dimensional plot of the predicted solution.
Noise ( σ )
Estimated α^
Estimated β^
Estimated V^
0%
1.4954
0.5032
1.0014
1%
1.4929
0.4997
0.9983
5%
1.4947
0.4961
0.9938
Table 8: Simultaneous Parameter Recovery Profile under Synthetic Noise
Figure 12: Convergence of Parameters for each noise level
Figure 13: Computational cost of each collocation methodology
Figure 14: Predicted vs exact solution on each edge at t=1s .
Fractional Scheme Used
L1
L2−1σ
Noise ( σ )
Estimated γ^
Estimated V^
Estimated γ^
Estimated V^
0%
0.4920
1.0003
0.4974
1.0010
1%
0.4918
0.9855
0.4979
0.9925
5%
0.4866
0.9611
0.4871
0.9680
Table 12: Simultaneous Parameter Recovery Profile under Synthetic Noise
Figure 15: Parameter convergence history.
Figure 21
Time
M^(t^)−M^(0)
Iγ[Q^net](t^)
Rel. Error (%)
t^=0.20
5.9106×10−1
5.9061×10−1
0.077
t^=0.50
8.6204×100
8.6190×100
0.016
t^=0.80
7.4173×100
7.3766×100
0.549
t^=1.00
4.0244×100
4.0561×100
0.790
Table 14: Open-System Mass Conservation, γ=0.85
Junction
Time
Q^in
Q^out
Rel. Residual (%)
E
t^=0.20
2.0283×10−2
2.1648×10−2
6.308
E
t^=0.50
1.6337×101
1.6341×101
0.026
E
t^=0.80
7.8457×100
7.8419×100
0.048
E
t^=1.00
2.0642×100
2.0688×100
0.224
F
t^=0.50
1.1534×101
1.1523×101
0.093
F
t^=0.80
1.4285×101
1.4284×101
0.006
Table 15: Junction Continuity Residuals, γ=0.85
Figure 16: (a) Water level profile along the path B→E→F→D at time t after reversing the normalization. (b) Heatmap representing the water height fluctuation at time t=5s .
Figure 17: IEEE 14-Bus System Topology (0-Indexed)
Figure 18: Plot of energy vs time in Log-Log axes
Metric / Quantity
γ=0.50
γ=0.75
γ=1.00
Initial-Boundary Conditions
Initial Pulse Relative RMS Error
0.388%
0.558%
0.467%
Initial Pulse L2 Error
4.206×10−4
6.038×10−4
5.057×10−4
Initial Velocity L2 Error
1.516×10−3
2.488×10−3
2.324×10−3
Network Transmission Constraints
Junction Continuity L2 Error
3.707×10−3
3.872×10−3
2.493×10−3
Table 17: Comprehensive Physical-Consistency and Energy Diagnostics Across Fractional Orders γ .
Figure 19: Propogation of the voltage surge at time t
Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.
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