Water temperature is known to be principal driver of the growth, survival, and reproduction of economically viable fish [21, 30] (see Appendix for more details). Increases in water temperature are also linked to the occurrence of aquatic invasive species [28, 29], which may displace fish and native aquatic organisms, and further result in harmful algal blooms [9, 26]. Hence, accurate and timely information about water temperature is necessary to monitor the ecological health of lakes and forecast future populations of fish and other aquatic taxa. Since observations of water temperatures are incomplete at broad spatial scales (or nonexistent for most lakes), physics-based models of lake temperature, e.g., the General Lake Model (GLM) [10], are commonly used for studying lake processes. A standard formulation in these models is to assume horizontal heterogeneity is limited and that the most relevant dynamics are captured in the vertical dimension of the lake, thereby modeling the lake as a series of vertical layers. These modeling studies often use temperature of water at the centre of a lake at varying depth values 1 and time points for model validation. We adopt the same formulation to model the temperature of water in a lake, Y d,tat depth d and time t . In particular, we leverage two key physical principles of our problem to guide neural network approaches, briefly described in the following.

This paper introduces a novel framework for combining scientific knowledge of physics-based models with neural networks to advance scientific discovery. This framework, termed as physics-guided neural network (PGNN), leverages the output of physics-based model simulations along with observational features to generate predictions using a neural network architecture. Further, this paper presents a novel framework for using physics-based loss functions in the learning objective of neural networks, to ensure that the model predictions not only show lower errors on the training set but are also scientifically consistent with the known physics on the unlabeled set. We illustrate the effectiveness of PGNN for the problem of lake temperature modeling, where physical relationships between the temperature, density, and depth of water are used to design a physics-based loss function. By using scientific knowledge to guide the construction and learning of neural networks, we are able to show that the proposed framework ensures better generalizability as well as scientific consistency of results.