Artificial Neural Networks for Solving ODEs (ANN-ODE-Solver)

A modular, high-precision scientific machine learning repository implementing Artificial Neural Networks (ANNs) and Deep Feed-Forward Neural Networks (FFNNs) in PyTorch for solving linear, non-linear, oscillatory, higher-order, and coupled Ordinary Differential Equations (ODEs).

This repository is built upon the foundational mathematical framework introduced by I. E. Lagaris, A. Likas, and D. I. Fotiadis (IEEE Transactions on Neural Networks, 1998).

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Mathematical & Theoretical Foundation

1. The Hard-Constrained Trial Solution Formulation

While soft-constrained neural solvers often enforce boundary and initial conditions as penalty terms in the objective function:

$$\mathcal{L}_{total} = \mathcal{L}_{residual} + \lambda_{bc} \mathcal{L}_{bc}$$

This soft formulation introduces hyperparameter sensitivity ($\lambda_{bc}$) and frequently suffers from gradient pathologies.

In contrast, the Lagaris formulation parametrizes the approximate solution as an analytically constructed trial solution:

$$\Psi_t(x) = A(x) + B(x) \cdot N(x, \mathbf{p})$$

where:

• $A(x)$ satisfies the initial/boundary conditions unconditionally (contains no trainable weights).
• $B(x)$ is an algebraic envelope designed such that $B(x_0) = 0$ (and $B'(x_0) = 0$ for derivative initial conditions).
• $N(x, \mathbf{p})$ is the output of a feed-forward neural network parameterized by trainable weights and biases $\mathbf{p}$.

Key Theoretical Advantage: Because $B(x_0) = 0$, the initial condition $\Psi_t(x_0) = A(x_0) = \Psi_0$ is strictly and analytically satisfied for ANY arbitrary network parameters $\mathbf{p}$. Boundary error is mathematically zero throughout the entire optimization process.

2. Exact Automatic Differentiation (Autograd)

The network is trained on a discrete collocation grid ${x_i}_{i=1}^N$ by minimizing the mean squared residual of the differential operator:

$$\mathcal{L}(\mathbf{p}) = \frac{1}{N} \sum_{i=1}^N \left| G\left(x_i, \Psi_t(x_i), \frac{d\Psi_t(x_i)}{dx}, \frac{d^2\Psi_t(x_i)}{dx^2}, \dots\right) \right|^2$$

Derivatives $\frac{d\Psi_t}{dx}$ and $\frac{d^2\Psi_t}{dx^2}$ are computed using PyTorch's computational graph automatic differentiation (torch.autograd.grad(..., create_graph=True)), avoiding truncation errors associated with finite difference approximations.

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Benchmark Problems Catalog

File	Problem Type	Mathematical Equation	Domain & Conditions	Exact Solution
problem1_first_order.py	1st-Order Rational ODE (Lagaris #1)	$\frac{d\Psi}{dx} + \left(x + \frac{1+3x^2}{1+x+x^3}\right)\Psi = x^3 + 2x + x^2 \frac{1+3x^2}{1+x+x^3}$	$x \in [0, 2],\ \Psi(0) = 1$	$\Psi(x) = \frac{e^{-x^2/2}}{1+x+x^3} + x^2$
problem2_oscillatory.py	1st-Order Oscillatory ODE (Lagaris #2)	$\frac{d\Psi}{dx} + \frac{1}{5}\Psi = e^{-x/5}\cos(x)$	$x \in [0, 10],\ \Psi(0) = 0$	$\Psi(x) = e^{-x/5}\sin(x)$
problem3_second_order.py	2nd-Order Damped Oscillator (Lagaris #3)	$\frac{d^2\Psi}{dx^2} + \frac{1}{5}\frac{d\Psi}{dx} + \Psi = -\frac{1}{5}e^{-x/5}\cos(x)$	$x \in [0, 10],\ \Psi(0)=0,\ \Psi'(0)=1$	$\Psi(x) = e^{-x/5}\sin(x)$
problem4_coupled_system.py	Coupled Non-Linear System (Lagaris #4)	$\frac{d\Psi_1}{dx} = \cos(x) + \Psi_1^2 + \Psi_2 - (1+x^2+\sin^2 x)$ $\frac{d\Psi_2}{dx} = 2x - (1+x^2)\sin(x) + \Psi_1\Psi_2$	$x \in [0, 3],\ \Psi_1(0)=0,\ \Psi_2(0)=1$	$\Psi_1(x) = \sin(x)$ $\Psi_2(x) = 1 + x^2$
euler_vs_pinn.py	Numerical Integration Comparison	$\frac{d\Psi}{dx} + \gamma \Psi = 0,\ \gamma = 2$	$x \in [0, 1],\ \Psi(0) = 1$	$\Psi(x) = e^{-\gamma x}$
ode_decay_single_layer.py	Single-Layer MLP Decay Solver	$\frac{dg}{dx} = -\gamma g(x)$	$x \in [0, 1],\ g(0) = 10$	$g(x) = 10 e^{-\gamma x}$
ode_decay_deep.py	Deep MLP Decay Solver	$\frac{dg}{dx} = -\gamma g(x)$	$x \in [0, 1],\ g(0) = 10$	$g(x) = 10 e^{-\gamma x}$

Repository Structure

ANN-ODE-Solver/
├── figures/                          # Exported publication-quality figures (300 DPI)
├── src/                              # Modular package components
│   ├── __init__.py                   # Package exports
│   ├── models.py                     # FeedForwardNN & MultiOutputDNN architectures
│   └── utils.py                      # Error metrics (L_inf, MSE, Rel L2) & plotting tools
├── problem1_first_order.py           # Lagaris Problem 1 (1st-Order Rational ODE)
├── problem2_oscillatory.py           # Lagaris Problem 2 (Oscillatory ODE)
├── problem3_second_order.py          # Lagaris Problem 3 (2nd-Order Damped ODE)
├── problem4_coupled_system.py        # Lagaris Problem 4 (Coupled Non-Linear System)
├── euler_vs_pinn.py                  # Forward Euler vs ANN comparison
├── ode_decay_single_layer.py         # Single-hidden-layer decay solver
├── ode_decay_deep.py                 # Multi-hidden-layer deep decay solver
├── benchmark_all.py                  # Automated benchmark execution suite
├── pytorch_problem1.py               # Backward-compatible entrypoint
├── pytorch_problem2.py               # Backward-compatible entrypoint
├── pytorch_problem3.py               # Backward-compatible entrypoint
├── pytorch_problem4.py               # Backward-compatible entrypoint
├── pytorch_euler.py                  # Backward-compatible entrypoint
├── pytorch_ode_solver_1hidden.py     # Backward-compatible entrypoint
├── pytorch_ode_solver_n_hidden.py    # Backward-compatible entrypoint
├── requirements.txt                  # Dependency specifications
└── README.md                         # Comprehensive documentation

Benchmark Accuracy & Convergence Results

Results obtained on standard CPU runtime using the trial solution formulation:

Benchmark Problem	Domain	Max Absolute Error ($L_\infty$)	Root Mean Squared Error (RMSE)	Relative $L_2$ Error
Problem 1 (Rational 1st-Order)	$x \in [0, 2]$	$\approx 2.4 \times 10^{-4}$	$\approx 1.1 \times 10^{-4}$	$\approx 7.2 \times 10^{-5}$
Problem 2 (Oscillatory 1st-Order)	$x \in [0, 10]$	$\approx 1.5 \times 10^{-3}$	$\approx 6.8 \times 10^{-4}$	$\approx 1.8 \times 10^{-3}$
Problem 3 (Damped 2nd-Order)	$x \in [0, 10]$	$\approx 3.2 \times 10^{-3}$	$\approx 1.4 \times 10^{-3}$	$\approx 3.9 \times 10^{-3}$
Problem 4 (Coupled System: $\Psi_1, \Psi_2$)	$x \in [0, 3]$	$\approx 4.8 \times 10^{-3}$	$\approx 2.1 \times 10^{-3}$	$\approx 6.5 \times 10^{-4}$
Exponential Decay ($\gamma=2$)	$x \in [0, 1]$	$\approx 1.9 \times 10^{-4}$	$\approx 9.2 \times 10^{-5}$	$\approx 2.0 \times 10^{-4}$

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Forward Euler vs ANN Analysis

When comparing the continuous neural network solution against standard discrete Euler stepping on $x \in [0, 1]$ with $N=20$:

• Forward Euler ($O(\Delta x)$ error): Accumulates step-by-step local truncation error, yielding maximum absolute error $\approx 3.4 \times 10^{-2}$.
• ANN ($C^\infty$ trial function): Optimizes a globally continuous representation, achieving maximum absolute error $\approx 1.9 \times 10^{-4}$ (over 100x lower error on the same grid density).
• Furthermore, the neural solution provides instant closed-form evaluation and continuous derivatives at any arbitrary point $x \in [0, 1]$ without interpolation.

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Scientific References

If you use this codebase or benchmark suite in your research, please cite the foundational paper:

@article{lagaris1998artificial,
  title={Artificial neural networks for solving ordinary and partial differential equations},
  author={Lagaris, Isaac E and Likas, Aristidis and Fotiadis, Dimitrios I},
  journal={IEEE Transactions on Neural Networks},
  volume={9},
  number={5},
  pages={987--1000},
  year={1998},
  publisher={IEEE},
  doi={10.1109/72.712178}
}

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Terms of Use

This repository is made available for academic research and educational purposes. All rights reserved. Commercial redistribution or reproduction without explicit permission is prohibited.