Lecture notes on Physics Informed Neural Networks, Neural Operators, and their applications
cs.LG, cs.AI, cs.CE, physics.comp-ph
Submitted: 2026-09-15
Updated: 2026-09-15
Comments: 264 pages; Lecture notes for the PhD course Physics Informed Neural Network held at the University of Bozen/Bolzano
Code: https://github.com/androbomb/PINN_Course_2026
License: http://creativecommons.org/licenses/by/4.0/
The gist: This is the set of lecture notes for the PhD course Physics Informed Neural Network, held at the University of Bozen/Bolzano in the academic year 2025/2026.
Terminology
Abstract
This is the set of lecture notes for the PhD course Physics Informed Neural Network, held at the University of Bozen/Bolzano in the academic year 2025/2026. The goal of the course was to introduce the concept of Physics Informed Deep Neural Networks (PINN) and Neural Operators (NOs), discuss their implementation from scratch in PyTorch and using advanced ad-hoc developed open-source libraries such as NVIDia PhysicsNeMo to address real-world problems in various fields (engineering, physics, petroleum reservoir). We discuss recent topics such as Mixture-of-Models, Fourier Neural Operators, Physics-Informed Kolmogorov-Arnold Networks (PIKANs) and Fourier Neural Operators.
Sources
- Variance Reduction in SGD by Distributed Importance Sampling
- Neural ODEs as the Deep Limit of ResNets with constant weights
- Residual-based attention in physics-informed neural networks
- Neural Poisson Surface Reconstruction: Resolution-Agnostic Shape Reconstruction from Point Clouds
- Neural Machine Translation by Jointly Learning to Align and Translate
- FourCastNet 3: A geometric approach to probabilistic machine-learning weather forecasting at scale
- Spectral Networks and Locally Connected Networks on Graphs
- ShapeNet: An Information-Rich 3D Model Repository
- Multi-Task Learning with Deep Neural Networks: A Survey
- Sobolev Training for Neural Networks
- Fourier Neural Operators Explained: A Practical Perspective
- An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale
- State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning
- A Review of Sparse Expert Models in Deep Learning
- Estimating Information Flow in Deep Neural Networks
- Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers
- The Cost-Accuracy Trade-Off In Operator Learning With Neural Networks
- Kolmogorov-Arnold Networks: A Critical Assessment of Claims, Performance, and Practical Viability
- SoftAdapt: Techniques for Adaptive Loss Weighting of Neural Networks with Multi-Part Loss Functions
- Physics-informed neural networks for inverse problems in supersonic flows
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