Inverse Methods in Heat Transfer

NPTEL-NOC IITM via YouTube

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Explore advanced computational techniques for solving inverse problems in heat transfer through this comprehensive course that bridges traditional mathematical methods with modern machine learning approaches. Begin with fundamental concepts of inverse problems, understanding how they differ from direct problems and examining the unique challenges they present, including ill-posedness and non-linearity. Master linear regression techniques and normal equations for solving inverse conduction problems, then progress to nonlinear inverse problems using gradient descent, Newton algorithms, and Gauss-Newton methods. Learn essential regularization techniques including Tikhonov regularization and Levenberg-Marquardt algorithms to handle overfitting and improve solution stability. Develop a solid foundation in probability theory, Bayes theorem, and statistical methods, applying maximum likelihood estimation, MAP estimates, and Bayesian inference to inverse heat transfer problems. Discover how to implement Metropolis-Hastings Markov Chain Monte Carlo (MHMCMC) methods for complex probabilistic inverse problems. Transition into machine learning fundamentals, exploring supervised learning, classification, and logistic regression as tools for inverse problem solving. Build and train neural networks from basic perceptrons to multi-layer networks, understanding forward propagation and backpropagation algorithms. Investigate cutting-edge Physics-Informed Neural Networks (PINNs) and their application to inverse heat transfer problems, learning how to incorporate boundary conditions and physical constraints into neural network architectures. Apply these methods through practical coding examples in MATLAB, working with real heat transfer scenarios including unsteady conduction problems, and develop both surrogate model-based and PINN-based inverse solutions for complex thermal systems.

Syllabus

#1 Introduction to the Inverse Methods in Heat Transfer Course
#2 Inverse Problems | Definition, History & Applications | Inverse Methods in Heat Transfer
#3 The inverse Problem Solving Process | Inverse Methods in Heat Transfer
#4 Review of Basic Heat Transfer for this Course | Inverse Methods in Heat Transfer
#5 Introduction To Week 02 | Inverse Methods in Heat Transfer
#6 Introduction to Linear Regression for Inverse Problems | Inverse Methods in Heat Transfer
#7 Example | Application of Linear Regression for an Inverse Conduction Problem
#8 Goodness of Fit & Coefficient of Determination | Inverse Methods in Heat Transfer
#9 Linear Regression with Quadratic Model | Inverse Methods in Heat Transfer
#10 Summary of Week 02 | Inverse Methods in Heat Transfer
#11 Introduction To Week 03 | Inverse Methods in Heat Transfer
#12 Introduction to Normal Equations for Linear Models | Part 1 | Inverse Methods in Heat Transfer
#13 Normal Equations for Linear Models | Part 2 | Inverse Methods in Heat Transfer
#14 Parity Plots | Inverse Methods in Heat Transfer
#15 Programming Inverse Methods using Normal Equations | Inverse Methods in Heat Transfer
#16 Variants on the Linear Model for Inverse Problems | Inverse Methods in Heat Transfer
#17 Summary Of Week 03 | Inverse Methods in Heat Transfer
#18 The General Inverse Methods Process | Inverse Methods in Heat Transfer
#19 Simple Nonlinear Inverse Problem | Transient Heat Transfer | Inverse Methods in Heat Transfer
#20 Review of Required Calculus Results | Inverse Methods in Heat Transfer
#21 Gradient Descent Algorithm | Inverse Methods in Heat Transfer
#22 Gradient Descent | Simple Example | Inverse Methods in Heat Transfer
#23 Gradient Descent for Nonlinear Inverse Problem | Theory | Inverse Methods in Heat Transfer
#24 Gradient Descent for Nonlinear Inverse Problem Coding Example Theory
#25 Newton Algorithm for a System of Equations | Inverse Methods in Heat Transfer
#26 Gauss Newton Algorithm | Derivation & Code | Inverse Methods in Heat Transfer
#27 Overfitting & Regularization for Linear Models | Inverse Methods in Heat Transfer
#28 Tikhonov Regularization & Levenberg Marquardt | Theory | Inverse Methods in Heat Transfer
#29 Tikhonov & Levenberg Marquardt | Example Code | Inverse Methods in Heat Transfer
#30 Introduction to Probability for Inverse Methods | Inverse Methods in Heat Transfer
#31 Sum & Product Rules of Probability | Inverse Methods in Heat Transfer
#32 Bayes Theorem | Simple Examples | Inverse Methods in Heat Transfer
#33 Independence & Expectation | Inverse Methods in Heat Transfer
#34 Variance & Covariance | Inverse Methods in Heat Transfer
#35 Gaussian Distribution | The Standard Normal Table | Inverse Methods in Heat Transfer
#36 Maximum Likelihood Estimate | Inverse Methods in Heat Transfer
#37 MLE, MAP Estimates | Inverse Methods in Heat Transfer
#38 Introduction to Bayesian Methods for Inverse Problems | Inverse Methods in Heat Transfer
#39 Offline Bayesian Estimation | Inverse Methods in Heat Transfer
#40 Offline Bayesian Estimation | MATLAB Demo | Inverse Methods in Heat Transfer
#41 MHMCMC for Inverse Problems | Inverse Methods in Heat Transfer
#42 MHMCMC for Inverse Problems | MATLAB Demo | Inverse Methods in Heat Transfer
#43 Why Machine Learning in Inverse Heat Transfer? | Inverse Methods in Heat Transfer
#44 Overview of AI & ML | Inverse Methods in Heat Transfer
#45 Supervised Machine Learning as an Inverse Problem | Inverse Methods in Heat Transfer
#46 Introduction to Week 9 | From Linear Models to Neural Networks | Inverse Methods in HeatTransfer
#47 Gradient Descent | Batch, Stochastic & Mini Batch | Inverse Methods in Heat Transfer
#48 Logistic Regression | The Forward Model | Inverse Methods in Heat Transfer
#49 Logistic Regression | Binary Entropy Cost Function & Gradient | Inverse Methods inHeat Transfer
#50 Multiclass Classification | Inverse Methods in Heat Transfer
#51 Linear Separability & Neural Networks | Inverse Methods in Heat Transfer
#52 Introduction to Week 10 | XOR & Deeper Networks | Inverse Methods in Heat Transfer
#53 Forward Pass Through a Simple Neural Network | Inverse Methods inHeat Transfer
#54 Backprop in a Scalar Chain | Inverse Methods in Heat Transfer
#55 Backprop in a MLP | Inverse Methods in Heat Transfer
#56 Introduction to Week 11 | ANNs as Surrogate Models | Inverse Methods in Heat Transfer
#57 Physics Informed Neural Networks | Introduction | Inverse Methods in Heat Transfer
#58 Physics Informed Neural Networks | An Intuitive Explanation | Inverse Methods in Heat Transfer
#59 Physics Informed Neural Networks | BC Incorporation | Inverse Methods in Heat Transfer
#60 PINNs for Inverse Problems | Inverse Methods in Heat Transfer
#61 Introduction to Week 12 | Sensitivity Analysis | Inverse Methods in Heat Transfer
#62 Code Examples of Logistic Regression | OR & AND Gates | Inverse Methods in Heat Transfer
#63 Code Example of Shallow Neural Network | XOR Gate | Inverse Methods in Heat Transfer
#64 Code Walkthrough for PINNs in Burgers Equation | Inverse Methods in Heat Transfer
#65 Formulation of a PINN Based Inverse Problem in Unsteady Conduction
#66 Formulation of a Surrogate Model Based Inverse Solution in Unsteady Conduction
#67 Summary of Course | Inverse Methods in Heat Transfer