The two-dimensional laser-based Angstrom method measures the in-plane thermal properties for anisotropic film-like materials. It involves periodic laser heating at the center of a suspended film sample and records its transient thermal response by infrared imaging. These spatiotemporal temperature data must be analyzed to extract the unknown thermal conductivity values in the orthotropic directions, an inverse parameter fitting problem. Previous development demonstration of the metrology used a least squares fitting method that relies on numerical differentiation to evaluate the second-order partial derivatives in the differential equation describing transient conduction in the physical system. This fitting approach is susceptible to measurement noise, introducing high uncertainty in the extracted properties when working with noisy data. For example, when noise of signal-to-noise ratio of 10 is added to simulated amplitude and phase data, the error in the extracted thermal conductivity can exceed 80 %. In this work, we introduce a new alternative inverse parameter fitting approach using physics-informed neural networks (PINNs) to increase the robustness of the measurement technique for noisy temperature data. We demonstrate the effectiveness of this approach even for scenarios with extreme levels of noise in the data. Specifically, the PINNs-approach accurately extracts the properties to within 5 % of the true values even for high noise levels (signal-to-noise ratio of 1). This offers a promising avenue for improving the robustness and accuracy of advanced thermal metrology tools that rely on inverse parameter fitting of temperature data to extract thermal properties