๐ฌ Advanced Scientific Computing with Deep Learning
B.Tech Final Year Project - Information Technology
Siddhant Manna | Meghnad Saha Institute of Technology | 2025
This comprehensive research project explores the revolutionary application of Physics-Informed Neural Networks (PINNs) for solving complex partial differential equations (PDEs). Unlike traditional numerical methods, PINNs embed physical laws directly into neural network architectures, creating a powerful framework that bridges deep learning and computational physics.
- Develop PINN frameworks for multiple PDE types
- Achieve high accuracy with minimal training data
- Demonstrate superiority over traditional numerical methods
- Explore advanced techniques like curriculum learning and Fourier embeddings
| Metric | Value | Metric | Value |
|---|---|---|---|
| ๐ Report Pages | 60+ | ๐ง PDEs Solved | 3 |
| ๐ป Training Time | 9 min 3 sec | ๐ References | 13+ |
| ๐ฏ Best L2 Error | 6.8ร10โปโด | ๐ง GPU Used | NVIDIA T4 |
| โก Max Iterations | 150,000 | ๐ Convergence | Achieved |
- Application: Fluid dynamics, shock wave modeling
- Achievement: L2 error of 6.8ร10โปโด
- Training Time: 9 minutes on NVIDIA T4
- Application: Phase separation, material science
- Key Feature: Multi-component alloy modeling
- Enhancement: Curriculum learning integration
- Application: Wave propagation, optimal path planning
- Innovation: Backward time integration
- Performance: Superior to fast-sweeping methods
Figure 1: Distribution of 10,000 collocation points, 50 initial conditions, and 50 boundary points for PINN training
Figure 2: PINN training loss convergence over 5,000 epochs with piecewise learning rate decay, achieving L2 error of 6.8ร10โปโด
Figure 3: 3D visualization of Burgers equation solution showing shock formation at t=0.4 and temporal evolution
Figure 4: Side-by-side comparison demonstrating PINN advantages over traditional numerical methods
- Hidden Layers: 8-9 layers with 20 neurons each
- Activation: Hyperbolic tangent (tanh)
- Normalization: Input scaling to [-1,1]
- Boundary Encoding: Strict Dirichlet condition adherence
- Progressive parameter complexity increase
- Stage-wise training approach
- Enhanced stability for complex PDEs
- Time-dependent accuracy enforcement
- Temporal causality preservation
- Improved generalization over time
- Spatial periodicity encoding
- Dramatic error reduction (2+ orders of magnitude)
- Enhanced spatial accuracy
| Equation | Method | L2 Error | Training Time |
|---|---|---|---|
| Burgers | PINN | 6.8ร10โปโด | 9 min 3 sec |
| Allen-Cahn | PINN+FFE | Significantly reduced | - |
| Eikonal | PINN+Causal | Superior accuracy | 150k iterations |
- PINNs vs Traditional: Higher accuracy with less computational resources
- FFE Enhancement: 2+ orders of magnitude error reduction
- Causal Training: Consistent convergence advantages
- GPU Acceleration: Optimal performance on NVIDIA T4
- GPU: NVIDIA T4 Tensor Core
- Platform: Google Colab
- Precision: Float32 optimization
- Memory: Optimized for large-scale problems
tensorflow >= 2.x # Deep learning framework
numpy # Numerical computations
matplotlib # Visualization
scipy # Scientific computing- Automatic Differentiation: TensorFlow GradientTape
- Optimization: Adam with adaptive learning rates
- Loss Functions: Multi-component physics-informed loss
- Sampling: Uniform and adaptive collocation strategies
- Novel PINN formulations for time-dependent PDEs
- Integration of curriculum learning with physics constraints
- Causal training methodology for temporal problems
- FFE enhancement for periodic boundary conditions
- Efficient GPU-accelerated training pipelines
- Boundary-encoded output layers
- Multi-stage training protocols
- Comprehensive error analysis frameworks
- Rigorous comparison with analytical solutions
- Performance benchmarking against traditional methods
- Scalability analysis across different problem sizes
- Robustness testing under various conditions
- Fluid Dynamics: Turbulence modeling, flow optimization
- Materials Science: Alloy design, phase prediction
- Geophysics: Seismic wave propagation, exploration
- Robotics: Path planning, navigation systems
- Graduate Research: Advanced numerical methods
- Computational Physics: Modern simulation techniques
- Machine Learning: Physics-informed AI development
- Engineering: Real-world problem solving
- Multi-GPU Training: Distributed computing support
- 3D Extensions: Complex geometry handling
- Real-time Inference: Optimized deployment
- Uncertainty Quantification: Bayesian extensions
- Hybrid Methods: Classical-neural combinations
- Anisotropic Media: Complex material properties
- Multi-Physics Coupling: Interdisciplinary problems
- Inverse Problems: Parameter identification
- Transfer Learning: Cross-domain applications
- Title: Solution of Partial Differential Equations Using Physics Informed Neural Network
- Author: Siddhant Manna (Roll: 14200222065, Reg: 221420120620)
- Supervisor: Assistant Professor Indrajit Das
- Institution: Meghnad Saha Institute of Technology
- Department: Information Technology
- Year: 2025
- Raissi et al. (2019): Foundational PINN methodology
- Karniadakis et al. (2021): Physics-informed machine learning
- Multiple domain-specific applications and enhancements
- โ High Accuracy: L2 error of 6.8ร10โปโด for Burgers equation
- โ Computational Efficiency: 9-minute training on single GPU
- โ Novel Techniques: FFE, curriculum learning, causal training
- โ Comprehensive Validation: Multiple PDE types solved
- โ Research Impact: Advancing scientific computing methods
pip install tensorflow numpy matplotlib scipy# Clone the repository
git clone <repository-url>
cd pinn-pde-solver
# Run the main PINN implementation
python Physics-Informed-Neural-Network.py
# Visualize results
python visualize_results.pyFor research collaboration, technical discussions, or project inquiries:
โญ If this research contributes to your work, please consider citing! โญ
Advancing the frontiers of computational physics through deep learning ๐