A systematic computational method to simulate and detect sub-surface flaws, through non-destructive transient thermography, in aluminium sheets and friction stir welded sheets is proposed. The proposed method relies on feature extraction methods and a data-driven machine learning modelling structure. In this work, we propose the use of a multi-layer perceptron feed-forward neural-network with feature extraction methods to improve the flaw-probing depth of transient thermography inspection. Furthermore, for the first time, we propose Thermographic Signal Linear Modelling (TSLM), a hyper-parameterfree feature extraction technique for transient thermography. The new feature extraction and modelling framework was tested with out-of-sample experimental transient thermography data and results show effectiveness in sub-surface flaw detection of up to 2.3 mm deep in aluminium sheets (99.8 % true positive rate, 92.1 % true negative rate) and up to 2.2 mm deep in friction stir welds (97.2 % true positive rate, 87.8 % true negative rate).