# Scientific Machine Learning — Course Portfolio

> A six-lecture graduate-level course on Scientific Machine Learning (SciML), bridging rigorous physics, applied mathematics, and modern deep learning.

## About This Course

This course is designed to be accessible to a broad audience with a background in calculus, linear algebra, and introductory machine learning. Each lecture follows four principles: clear and concise mathematics, strong motivation, readable high-quality code, and insightful visualizations.

## Course Structure

### Lecture 1: Physics-Informed Neural Networks (PINNs)
- URL: /course/lecture-1-pinns
- Key concept: Embedding PDE residuals directly into the neural network loss function
- Benchmark: Burgers equation (ν = 0.01/π), relative L2 error ~5%
- Key paper: Raissi, Perdikaris, Karniadakis (2019). "Physics-informed neural networks." Journal of Computational Physics, 378, 686–707. DOI: 10.1016/j.jcp.2018.10.045
- Code: PyTorch implementation with automatic differentiation for PDE residuals

### Lecture 2: Neural Operators
- URL: /course/lecture-2-neural-operators
- Key concepts: DeepONet (Branch-Trunk architecture), Fourier Neural Operator (FNO)
- Benchmark: Antiderivative operator learning, relative L2 error ~2%
- Key papers:
  - Lu et al. (2021). "Learning nonlinear operators via DeepONet." Nature Machine Intelligence. DOI: 10.1038/s42256-021-00302-5
  - Li et al. (2021). "Fourier neural operator for parametric PDEs." ICLR 2021. arXiv:2010.08895
- Code: PyTorch implementations of DeepONet and FNO1d

### Lecture 3: World Models and JEPA
- URL: /course/lecture-3-world-models-jepa
- Key concepts: Joint Embedding Predictive Architecture, EMA target encoder, collapse prevention
- Benchmark: Van der Pol oscillator latent representation learning
- Key papers:
  - LeCun (2022). "A Path Towards Autonomous Machine Intelligence." OpenReview.
  - Assran et al. (2023). "I-JEPA." CVPR 2023. arXiv:2301.08243
  - Bardes et al. (2024). "V-JEPA." ICLR 2024.
- Code: PyTorch JEPA with EMA, stop-gradient, and PCA visualization

### Lecture 4: Generative Modeling and UQ in Chaotic Systems
- URL: /course/lecture-4-generative-uq
- Key concepts: Score-based diffusion (VP-SDE), denoising score matching, CRPS, ensemble spread-skill
- Benchmark: Lorenz-96 (40-dimensional chaotic system), probabilistic forecasting
- Key papers:
  - Song et al. (2021). "Score-Based Generative Modeling through SDEs." NeurIPS 2021. arXiv:2011.13456
  - Price et al. (2025). "GenCast." Nature. arXiv:2312.15796
- Code: PyTorch VP-SDE diffusion model with DDPM sampler and CRPS evaluation

### Lecture 5: Neural ODEs and Geometric Mechanics
- URL: /course/lecture-5-neural-odes-geometric-mechanics
- Key concepts: Neural ODEs (adjoint method), Hamiltonian Neural Networks, SymODEN
- Benchmark: Nonlinear pendulum, energy conservation over long rollouts
- Key papers:
  - Chen et al. (2018). "Neural Ordinary Differential Equations." NeurIPS. arXiv:1806.07366
  - Greydanus et al. (2019). "Hamiltonian Neural Networks." NeurIPS. arXiv:1906.01563
  - Zhong et al. (2020). "Symplectic ODE-Net." ICLR. arXiv:1909.12077
- Code: PyTorch comparison of Baseline NODE vs. HNN vs. SymODEN on pendulum

### Lecture 6: Engineering and Performance of SciML
- URL: /course/lecture-6-engineering-performance
- Key concepts: TFLOP counting, Roofline model, arithmetic intensity, precision strategy, kernel fusion, distributed training
- Key findings:
  - PINNs: memory-bound due to autograd; require float64 for stable training
  - FNO: memory-bandwidth limited; 2.5x speedup via fused FFT-GEMM-iFFT Triton kernels
  - Neural ODEs: compute-bound; benefit from JAX/XLA compilation
- Key paper: "High-Performance Fourier Neural Operator." arXiv:2504.11681
- Code: TFLOP profiler, Roofline model visualizer, precision benchmark

## Key Themes Across the Course

1. **Physics as Inductive Bias**: From hard constraints (SymODEN) to soft penalties (PINNs) to architectural priors (FNO)
2. **The Operator Perspective**: Learning mappings between function spaces, not just finite-dimensional vectors
3. **Uncertainty is Physical**: Chaotic systems require probabilistic forecasting; diffusion models provide principled UQ
4. **Structure Preservation**: Conservation laws must be respected by construction, not hoped for
5. **Engineering Matters**: Theoretical models must be profiled, optimized, and deployed on real hardware

## Machine-Readable Metadata

- Format: HTML5 with semantic tags (main, article, section, header, aside)
- Math: MathJax 3 (TeX-SVG renderer)
- Schema: JSON-LD (Course, CourseInstance, Person)
- Code: syntax-highlighted with language annotations
- Citations: BibTeX and APA formats available on each lesson page