###### Abstract  
In engineering design, oftentimes a system's dynamic response is known or can be measured, but the source generating these responses is not known. The mathematical problem where the focus is on inferring the source terms of the governing equations from the set of observations is known as an inverse source problem (ISP). ISPs are traditionally solved by optimization techniques with regularization, but in the past few years, there has been a lot of interest in approaching these problems from a deep-learning viewpoint. In this paper, we propose a deep learning approach--infused with physics information--to recover the forcing function (source term) of systems with one degree of freedom from the response data. We test our architecture first to recover smooth forcing functions, and later functions involving abruptly changing gradient and jump discontinuities in the case of a linear system. Finally, we recover the harmonic, the sum of two harmonics, and the gaussian function, in the case of a non-linear system. The results obtained are promising and demonstrate the efficacy of this approach in recovering the forcing functions from the data. physics informed neural network; dynamic force identification; deep learning; duffing's equation; spring mass damper system; non-linear oscillators +
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In the past few decades, plenty of research has been published discussing various approaches to solving this problem, and it will be difficult to enumerate them all given the scope of this paper; nonetheless, a few notable mentions are as follows. [PERSON] [9] used the conjugate gradient method to estimate the time-dependent forces in a non-linear oscillator. [PERSON] et al. [10] developed a recursive estimator based on the Kalman filter to determine the impulsive load from the measurement data for the single and multi-degree-of-freedom systems. In another interesting work by [PERSON] et al. [11], the authors proposed a dual Kalman filter for estimating the full states of a linear multi-degree of freedom system with unknown input from a limited number of noisy acceleration measurements and a known physical model. [PERSON]. [12] formulated the force identification problem of the duffing oscillator as a Volterra-type integral equation of the first kind and used the regularization technique to stabilize the solution. [PERSON] [13] proposed a method for predicting forces only from response data without the need for any parametric or governing equation information using the Hilbert transform. [PERSON]. [14] solved the non-linear force identification problem in the frequency domain using ordinary least squares with Tikhonov regularization and its variants. [PERSON] et al. [15] solved the non-linear vibration problem by transforming the non-linear ordinary differential equations into parabolic ordinary differential equations due to their robustness against large noise. Recently, [PERSON] et al. [16] proposed a calibration-based integral formulation for estimating the forcing function in the spring mass damper system from response data. For a detailed review of past and present literature on dynamic load identification techniques, interested readers are advised to refer to [17]. In recent years, there has been a significant interest in applying machine learning and deep learning techniques for load identification. [PERSON] and [PERSON] [18] proposed a technique for recovering the input forces from acceleration time history using dynamic principal component analysis. [PERSON] et al. [19] used a deep Recurrent Neural Networks (RNNs) technique with two variants of Long Short Term Memory (LSTM) to recover the impact loads on non-linear structures from response data. They tested their architecture on a damped duffing oscillator subjected to an impact load expressed by normal distribution function and on a composite plate.