🌊 Welcome Back

Scriber Labs develops educational and research-oriented computational models for exploring how structure and dynmaics can be recovered from data.

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🔖 Core Philosophical Commitments

1. Interpretiblity
2. Clarity (physics first, data second)
3. Parsimony
4. Scalability
5. Computationally efficient
6. Practicality (optional for low-fidelity projects and apps/widgets)
   • scientific value
   • educational
7. Sing-develepor feasibility

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🪀 Project 1: lf-pinn-harmonic-oscillator

🏡 Take-Home Messages:

Physics residual minimization does got guarantee physical inveriants unless sampling resolves the solution spectrum. This is a manifestation of spectral bias.

Residual minimization approximates the operator constraint, but conservation emerges from the generator s tructure. If the generator isnt structurally preserved (via sampling or architecture), invariants drift even under converged optimization.

• This connects spectral bias in neural nets to...
  • Nyquist sampling theory
  • Hamiltonian structure
  • Conservation Laws
  • PINN failure modes

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References

@article{basir2022pinnfailures,
  author    = {Basir, Soroush and Senocak, Inanc},
  title     = {Critical Investigation of Failure Modes in Physics-Informed Neural Networks},
  booktitle = {AIAA SCITECH 2022 Forum},
  year      = {2022},
  doi       = {10.2514/6.2022-2353}
}

@book{brunton2022datadriven,
  author    = {Brunton, Steven L. and Kutz, J. Nathan},
  title     = {Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control},
  publisher = {Cambridge University Press},
  year      = {2022},
  edition   = {2}
}

@incollection{hestenes1993hamiltonian,
  author    = {Hestenes, David},
  title     = {Hamiltonian Mechanics with Geometric Calculus},
  booktitle = {Clifford Algebras and Their Applications in Mathematical Physics},
  pages     = {203--214},
  publisher = {Springer},
  year      = {1993},
  doi       = {10.1007/978-94-011-1719-7_25}
}

@article{kutzbrunton2022parsimony,
  author  = {Kutz, J. Nathan and Brunton, Steven L.},
  title   = {Parsimony as the ultimate regularizer for physics-informed machine learning},
  journal = {Nonlinear Dynamics},
  volume  = {107},
  number  = {3},
  pages   = {1801--1817},
  year    = {2022},
  doi     = {10.1007/s11071-021-07118-3}
}

@article{raissi2019pinns,
  author  = {Raissi, Maziar and Perdikaris, Paris and Karniadakis, George Em},
  title   = {Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations},
  journal = {Journal of Computational Physics},
  volume  = {378},
  pages   = {686--707},
  year    = {2019},
  doi     = {10.1016/j.jcp.2018.10.045}
}