PINN vs VPINN for Steady-State Heat Conduction with Nonlocal Integral Boundary Condition

A comparative study of Physics-Informed Neural Networks (PINN) and Variational Physics-Informed Neural Networks (VPINN) applied to elliptic PDEs with a nonlocal integral boundary condition — in 1D and 2D, covering both linear and nonlinear (temperature-dependent conductivity) regimes.

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Problem Statement

We consider steady-state heat conduction with two formulations, in 1D and 2D:

1D Formulation

Linear: $$-u''(x) = f(x), \quad x \in (0, 1)$$

Nonlinear (temperature-dependent conductivity): $$-\bigl[k(u), u'(x)\bigr]' = f(x), \quad x \in (0, 1), \qquad k(u) = 1 + \beta, u^2$$

2D Formulation

Linear (Poisson): $$-\Delta u = f(x,y), \quad (x,y) \in (0, 1)^2$$

Nonlinear: $$-\nabla \cdot \bigl[k(u), \nabla u\bigr] = f(x,y), \quad (x,y) \in (0, 1)^2, \qquad k(u) = 1 + \beta, u^2$$

Boundary Conditions

All cases share a nonlocal integral boundary condition on the left edge and Dirichlet conditions elsewhere:

Boundary	1D	2D
Left	$u(0) = \alpha \int_0^1 u(x),dx + g_0$	$u(0,y) = \alpha \int_0^1 u(x,y),dx$ for each $y$
Right	$u(1) = g_1$	$u(1,y) = 0$
Bottom / Top	—	$u(x,0) = u(x,1) = 0$

This class of nonlocal conditions arises in models involving energy specification constraints, distributed sensors, and thermoelastic feedback systems.

Manufactured Solutions

1D: $u(x) = \sin(\pi x) + (1 - x) \cdot \frac{4}{\pi}$

2D: $u(x,y) = \sin(\pi y) \cdot \bigl[\sin(\pi x) + (1 - x) \cdot \frac{4}{\pi}\bigr]$

The 2D solution is a separable extension of the 1D case. The source term $f$ is computed by substitution into the governing equation.

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Methods

PINN — Strong Formulation

The PDE residual is enforced point-wise at interior collocation points. For nonlinear cases, the residual is computed via automatic differentiation on the flux $\boldsymbol{\varphi} = k(u_\theta), \nabla u_\theta$:

$$r_f(\mathbf{x}_i) = -\nabla \cdot \boldsymbol{\varphi}(\mathbf{x}_i) - f(\mathbf{x}_i)$$

This flux-based AD approach avoids manually expanding the chain rule.

VPINN — Weak (Variational) Formulation

The weak form is obtained by integration by parts against test functions:

• 1D: Legendre test functions $v_k(x) = P_k(2x-1) - 1$ with $v_k(1) = 0$
• 2D: Tensor product $v_i(x) \cdot w_j(y)$ where $v_i$ are Legendre and $w_j(y) = \sin(j\pi y)$

The boundary flux appears naturally as a trainable Lagrange multiplier — scalar $\lambda$ in 1D, vector $\boldsymbol{\lambda}$ (Fourier-sine coefficients) in 2D. Key advantages:

• Reduces the required differentiation order from 2 to 1
• Naturally accommodates the global integral constraint
• Provides the boundary flux as a free output

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Results

1D — Linear Case

Metric	PINN	VPINN
L2 error	8.74 × 10⁻⁶	3.49 × 10⁻⁶
L∞ error	1.84 × 10⁻⁵	6.58 × 10⁻⁶
H1 error	1.57 × 10⁻⁴	7.53 × 10⁻⁵

1D — Nonlinear Case ($\beta = 1$)

Metric	PINN	VPINN
L2 error	1.92 × 10⁻⁴	2.39 × 10⁻⁵
L∞ error	3.40 × 10⁻⁴	4.57 × 10⁻⁵
Integral BC	1.58 × 10⁻⁴	2.32 × 10⁻⁶

2D — Linear Case (Poisson)

Metric	PINN	VPINN
L2 error	3.09 × 10⁻⁵	1.79 × 10⁻³
Time (min)	18	31
$\lambda_0$	—	−1.8680 (exact: −1.8684)

VPINN improves to L2 = 1.92 × 10⁻⁴ with enriched test space ($N_{\text{test}} = 25 \times 8$).

2D — Nonlinear Case ($\beta = 1$)

Metric	PINN	VPINN
L2 error	7.14 × 10⁻⁵	5.16 × 10⁻⁴
$\lambda_0$	—	−4.1411 (exact: −4.1400)

Key Finding: Crossover Between VPINN and PINN in 2D

In 2D with sufficient test functions ($25 \times 8$), the VPINN surpasses the PINN for moderate nonlinearity ($\beta \leq 2.5$), but the PINN dominates at strong nonlinearity ($\beta \geq 3$). The crossover occurs around $\beta^* \approx 2.7$:

$\beta$	PINN L2	VPINN L2 (25×8)	Winner
0.5	1.54 × 10⁻⁴	1.23 × 10⁻⁴	VPINN
2.0	8.08 × 10⁻⁴	3.30 × 10⁻⁴	VPINN ×2.4
3.0	6.50 × 10⁻⁴	2.22 × 10⁻³	PINN ×3.4
5.0	7.13 × 10⁻⁴	1.71 × 10⁻²	PINN ×24

1D vs 2D Summary

Dimension	Regime	PINN L2	VPINN L2	Best default
1D	Linear	8.74 × 10⁻⁶	3.49 × 10⁻⁶	VPINN
1D	NL $\beta=1$	1.92 × 10⁻⁴	2.39 × 10⁻⁵	VPINN
2D	Linear	3.09 × 10⁻⁵	1.79 × 10⁻³	PINN
2D	NL $\beta=1$	7.14 × 10⁻⁵	5.16 × 10⁻⁴	PINN

In 1D the VPINN dominates. In 2D the PINN leads with default settings, but the VPINN catches up with enriched test spaces and surpasses PINN for moderate nonlinearity.

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Repository Structure

.
├── README.md
├── LICENSE
├── linear/                          # 1D linear (-u'' = f)
│   ├── requirements.txt
│   ├── src/
│   │   ├── exact_solution.py
│   │   ├── network.py
│   │   ├── utils.py
│   │   ├── pinn_solver.py
│   │   ├── vpinn_solver.py
│   │   └── run_all.py               # 4 studies + 5 figures
│   └── figures/
├── nonlinear/                       # 1D nonlinear (-[k(u)u']' = f)
│   ├── requirements.txt
│   ├── src/
│   │   ├── exact_solution.py
│   │   ├── network.py
│   │   ├── utils.py
│   │   ├── pinn_solver.py
│   │   ├── vpinn_solver.py
│   │   └── run_all.py               # 5 studies + 5 figures
│   └── figures/
├── 2D_linear/                       # 2D Poisson (-Δu = f)
│   ├── requirements.txt
│   ├── src/
│   │   ├── exact_solution.py
│   │   ├── network.py
│   │   ├── utils.py
│   │   ├── pinn_solver.py
│   │   ├── vpinn_solver.py
│   │   └── run_all.py               # 4 studies + 5 figures
│   └── results/                     # Figures + summary
├── 2D_nonlinear/                    # 2D nonlinear (-div(k∇u) = f)
│   ├── requirements.txt
│   ├── src/
│   │   ├── exact_solution.py
│   │   ├── network.py
│   │   ├── utils.py
│   │   ├── pinn_solver.py
│   │   ├── vpinn_solver.py
│   │   ├── run_all.py               # 5 studies + 5 figures
│   │   ├── study_ntest_beta_heatmap.py   # N_test × β analysis
│   │   └── study_deep_analysis.py        # Crossover + push N_test
│   └── results/                     # 10 figures + summaries

Quick Start

Requirements

• Python >= 3.10
• PyTorch >= 2.0
• NumPy >= 1.24
• SciPy >= 1.10 (for 2D FDM sparse solvers)
• Matplotlib >= 3.7

pip install torch numpy scipy matplotlib

Run All Benchmarks

# 1D linear (~30 min)
cd linear/src && python run_all.py

# 1D nonlinear (~1h)
cd nonlinear/src && python run_all.py

# 2D linear (~2.5h, CPU)
cd 2D_linear/src && python run_all.py

# 2D nonlinear (~4.5h, CPU)
cd 2D_nonlinear/src && python run_all.py

# Deep analysis: crossover + N_test push (~3h)
cd 2D_nonlinear/src && python study_ntest_beta_heatmap.py
cd 2D_nonlinear/src && python study_deep_analysis.py

Custom Configuration

All solvers accept a configuration dictionary :

from pinn_solver import train_pinn

results = train_pinn(config={
    "n_hidden": 64,
    "n_layers": 5,
    "beta": 2.0,       # nonlinearity strength (nonlinear cases)
    "n_adam": 20_000,
    "seed": 123,
})

print(f"L2 error: {results['errors']['L2']:.4e}")

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Key References

1. Raissi, M., Perdikaris, P. & Karniadakis, G.E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys., 378, 686-707.

2. Kharazmi, E., Zhang, Z. & Karniadakis, G.E. (2021). hp-VPINNs: Variational physics-informed neural networks with domain decomposition. CMAME, 374, 113547.

3. Liu, Y. (1999). Numerical solution of the heat equation with nonlocal boundary conditions. J. Comput. Appl. Math., 110(1), 115-127.

4. Cannon, J.R. (1963). The solution of the heat equation subject to the specification of energy. Quart. Appl. Math., 21(2), 155-160.

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Citation

If you use this code in your research or projects, please cite:

@software{auger2026pinn_integral_bc,
  author       = {Auger, Maxime},
  title        = {{PINN vs VPINN for Steady-State Heat Conduction with
                   Nonlocal Integral Boundary Condition (1D and 2D)}},
  year         = {2026},
  url          = {https://github.com/MaximeAuger/steady-state-heat-conduction},
  institution  = {FEMTO-ST Institute, Dept. of Applied Mechanics, SUPMICROTECH}
}

License

This project is licensed under the MIT License — see LICENSE for details.

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Author: Maxime Auger — FEMTO-ST Institute, Dept. of Applied Mechanics, ENSMM, Besancon, France.