Haptic feedback to the surgeon during robotic surgery would enable safer and more immersive surgeries but estimating tissue interaction forces at the tips of robotically controlled surgical instruments has proven challenging. Few existing surgical robots can measure interaction forces directly and the additional sensor may limit the life of instruments. We present a hybrid model and learning-based framework for force estimation for the Patient Side Manipulators (PSM) of a da Vinci Research Kit (dVRK). The model-based component identifies the dynamic parameters of the robot and estimates free-space joint torque, while the learning-based component compensates for environmental factors, such as the additional torque caused by trocar interaction between the PSM instrument and the patient's body wall. We evaluate our method in an abdominal phantom and achieve an error in force estimation of under 10% normalized root-mean-squared error. We show that by using a model-based method to perform dynamics identification, we reduce reliance on the training data covering the entire workspace. Although originally developed for the dVRK, the proposed method is a generalizable framework for other compliant surgical robots. The code is available at https://github.com/vu-maple-lab/dvrk_force_estimation.