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Physics-Guided Architecture (PGA) of Neural Networks for Quantifying Uncertainty in Lake Temperature Modeling
Arka Daw, R. Quinn Thomas, Cayelan C. Carey, Jordan S. Read, Alison P. Appling, Anuj Karpatne
TL;DR
Scientific deep learning needs to quantify uncertainty without sacrificing consistency with physical principles, but black-box models and dropout can produce physically inconsistent outputs. The paper proposes a physics-guided architecture for lake-temperature modeling and evaluates it on two lakes. PGA-LSTM produces physically consistent Monte Carlo samples, with a reported mean-sample test RMSE of 1.88°.
Problem
Scientific deep learning must provide uncertainty estimates while respecting physical principles, but black-box models and dropout can yield physically inconsistent outputs, especially when labeled data are scarce.
Source: §1 · p. 1Method
PGA-LSTM hard-wires lake physics through physics-informed neuron connections and physically meaningful intermediate variables, including a monotonicity-preserving module for predicting density.
Source: §1 · p. 2, §7 · p. 10Results
1.88° Mean Test RMSE was reported for the mean of PGA-LSTM's MC samples, alongside near-zero physical inconsistency and physically consistent solutions under MC-dropout perturbations.
Source: §5.1 · p. 8, §7 · p. 10Takeaways & Limitations
Near-zero physical inconsistency makes the MC samples meaningful for subsequent analyses of lake processes affected by temperature, such as fish growth and survival.
Source: §5.1 · p. 8Future work remains to test other state-of-the-art uncertainty methods and extend PGA to more complex spatial and temporal physical relationships.
Source: §7 · p. 10
Abstract
from arXiv · showhide
To simultaneously address the rising need of expressing uncertainties in deep learning models along with producing model outputs which are consistent with the known scientific knowledge, we propose a novel physics-guided architecture (PGA) of neural networks in the context of lake temperature modeling where the physical constraints are hard coded in the neural network architecture. This allows us to integrate such models with state of the art uncertainty estimation approaches such as Monte Carlo (MC) Dropout without sacrificing the physical consistency of our results. We demonstrate the effectiveness of our approach in ensuring better generalizability as well as physical consistency in MC estimates over data collected from Lake Mendota in Wisconsin and Falling Creek Reservoir in Virginia, even with limited training data. We further show that our MC estimates correctly match the distribution of ground-truth observations, thus making the PGA paradigm amenable to physically grounded uncertainty quantification.
1 Introduction
Scientific applications need deep-learning models that provide uncertainty estimates while respecting physical principles, a challenge heightened by scarce labeled data and dropout perturbations. This paper introduces a physics-guided architecture for lake-temperature modeling and compares it with baselines on two lakes.
1 Introduction
- Scientific applications may need uncertainty bounds alongside point estimates, and MC Dropout is presented as an uncertainty-quantification approach.The introduction highlights climate-change applications as one example.Source: §1 · p. 1
- Black-box models can produce spurious, physically inconsistent solutions when labeled training data are scarce, and dropout removes predictors and hidden states from calculations.The introduction identifies this tension between data-driven modeling and physical realism as a barrier to scientific applications.Source: §1 · p. 1
- MC dropout can disrupt physical consistency learned through physics-based losses, producing physically non-meaningful uncertainty estimates.PGL constrains outputs during training but leaves the neural-network architecture black-box, making it vulnerable to weight perturbations.Source: §1 · p. 2
- PGA embeds lake-temperature physics through physics-informed neuron connections and physically meaningful intermediate variables.The architecture is evaluated against baselines using data from Lake Mendota and Falling Creek Reservoir, which differ in physical characteristics and climatic regimes.Source: §1 · p. 2
2 Background and Related Work
Lake temperature models use vertical structure and water-density relationships as physical guides, while physics-guided learning adds physical-violation losses to prediction objectives. The section motivates physics-guided architectures for uncertainty estimation because dropout perturbations can undermine physical consistency in black-box networks.
2.1 Lake Temperature Modeling:
- Lake temperature modeling commonly represents a lake as vertical layers, and accurate temperature information matters for ecological health and fish populations.The paper adopts this vertical formulation for temperature at different depths and times.Source: §2.1 · p. 2
- Water is maximally dense at 4°C, and its density increases or remains constant with depth; these relationships guide the proposed architecture.The paper uses temperature-to-density mapping and monotonic density with depth as its two physical relationships.Source: §2.1 · pp. 2–3
2.2 Physics-guided Machine Learning:
- Physics-guided learning combines prediction loss with penalties for physical violations, constraining neural-network weights toward physically consistent options.Earlier lake-temperature work used these losses for temperature-density and density-depth relationships, with later LSTM work adding energy conservation [16] [13].Source: §2.2 · p. 3
- Physics-guided learning leaves architecture choices uninformed by physics and cannot guarantee consistent predictions on unseen instances; prior architectures encode principles not directly applicable here.The lake-temperature problem requires monotonic relationships and intermediate variables.Source: §2.2 · p. 3
2.3 Uncertainty
- Uncertainty quantification represents possible temperature values as a distribution, and the paper uses Monte Carlo dropout to sample predictions at test time.The paper notes that other Bayesian deep-learning approaches generally run more slowly than MC dropout, while PGA can be coupled with other UQ methods.Source: §2.3 · p. 3
- Dropout perturbs trained networks, and those perturbations can undo learned physical consistency in black-box architectures; embedding physics in the architecture improves the chance of consistent samples.The stated concern is that even small randomness from dropout may unlearn consistency acquired through physics-guided learning.Source: §2.3 · p. 3
3 Proposed Framework
PGA-LSTM combines temporal feature extraction, a monotonicity-preserving density predictor, and a density-to-temperature mapper. It trains the complete model using temperature and density losses with regularization.
3 Proposed Framework
- PGA-LSTM extracts temporal features, predicts density constrained to be nondecreasing with depth, then combines density and input drivers to predict temperature.The framework has an LSTM auto-encoder, a monotonicity-preserving LSTM, and a multilayer perceptron.Source: §3.1 · p. 4
3.2 Temporal Feature Extraction:
- An encoder maps the preceding seven days of feature vectors and the target-date vector to a compact representation that a decoder uses to reconstruct the sequence.The representation is intended to retain the sequential information of the previous week.Source: §3.2 · p. 4
3.3 Monotonicity-preserving
- The monotonicity-preserving LSTM tracks density alongside hidden and cell states, predicting positive density increments to enforce the known depth relationship.Its monotonicity-preserving design is also proposed for applications where target variables obey monotonic constraints.Source: §3.3 · pp. 4–5
3.4 Mapping Density to Temperature:
- Because density maps non-uniquely to temperature around 4°C, the model learns temperature from concatenated input drivers and predicted density.A small number of hidden layers is used because predicted density is already a strong physical predictor of temperature.Source: §3.4 · p. 5
3.5 End-to-end Learning Procedure:
- End-to-end training minimizes empirical losses for both temperature and density, plus a regularization term over the model weights.The density targets are obtained from observed temperatures using the one-to-one temperature-to-density mapping; λZ and λR weight density loss and regularization, respectively.Source: §3.5 · p. 5
4 Evaluation Setup
The evaluation uses data from Lake Mendota and Falling Creek Reservoir, with training and testing separated in time. It specifies the model, evaluation metrics, and LSTM-based comparison baselines.
4.1 Data and Experiment Design:
- Lake Mendota (35,213 observations) and Falling Creek Reservoir (7,588) provide contrasting datasets; the first four years train models, with later contiguous periods reserved for testing.Both lakes use meteorological drivers, depth, and simulated GLM temperature outputs as features; training subsets use randomly selected dates.Source: §4.1 · pp. 5–6
4.2 Model Specifications
- With seven-day histories, the monotonicity-preserving LSTM predicts density before a dense mapper predicts temperature; test-time dropout creates 100 networks per input.The dropout probability is 0.2.Source: §4.2 · p. 6
4.3 Evaluation Metrics:
- RMSE measures test-set generalizability, while Physical Inconsistency is the fraction of MC samples violating the density-depth relationship across consecutive depths.A tolerance of 10^-5 kg/m^3 determines whether consecutive-depth density differences count as inconsistent.Source: §4.3 · p. 6
4.4 Baselines:
- The baselines are a similarly structured black-box LSTM and a PGL-LSTM using physics-based loss; PGRNN is excluded because it models different physics and recurrence dimensions.Tables 1 and 2 report results on Lake Mendota and Falling Creek Reservoir, respectively, using 40% training data.Source: §4.4 · pp. 6–7, Table 1 · p. 7, Table 2 · p. 7
5 Results
PGA-LSTM achieves the lowest per-sample RMSE while preserving physical consistency under MC dropout, and remains strongest as training data shrink. Its per-sample predictions also avoid the physically inconsistent samples that can make baseline uncertainty estimates unusable.
5.1 Comparing PGA-LSTM with Baselines:
- LSTM MC samples are physically inconsistent 32% of the time, and averaging can lower RMSE without resolving high physical inconsistency.The authors caution that physically non-meaningful samples cannot support subsequent scientific analyses, including uncertainty quantification.Source: §5.1 · p. 7
- On FCR, PGA-LSTM lowers Per Sample Test RMSE from 2.96° for LSTM to 2.19° while keeping Physical Inconsistency close to 0; its Mean Test RMSE is 1.88°.The paper reports similar baseline-comparison trends on Lake Mendota and FCR; Table 1 and Table 2 compare both metrics using 40% training data.Source: §5.1 · pp. 7–8
5.2 Effect of Varying Training Size:
- As training fractions decrease, Test RMSE rises for all methods, but PGA-LSTM has the lowest Test RMSE at every fraction on both lakes.On FCR, its RMSE rises slowest as training data are reduced.Source: §5.2 · p. 8, Figure 6 · p. 8
6 Analysis of Results
Temperature-profile visualizations show that PGA-LSTM maintains physical consistency and its MC samples encompass ground-truth observations across lake depths. Its uncertainty estimates are slightly wider than ideal, while baseline estimates are slightly over-confident.
6.1 Visualizing Temperature Profiles:
- PGA-LSTM’s sample temperature profiles remain physically consistent, unlike the highly inconsistent LSTM and PGL-LSTM profiles, even when those baselines appear close to observations.The comparison uses 15 randomly selected dropout samples from a representative test date.Source: §6.1 · p. 8, Figure 7 · p. 9
- PGA-LSTM’s sample distribution envelops ground-truth observations at every depth, while baseline distributions fail to contain observations in deeper water.The plotted mean and variance use the complete dropout-sample pool across 10 training runs; the error bars represent two standard deviations, capturing 99.7% under a Gaussian assumption.Source: §6.1 · pp. 8–9, Figure 7 · p. 9
- PGA-LSTM sample variance gradually increases with depth, where missing water-clarity inputs make temperatures harder to predict on unseen years.The authors interpret the greater deep-water uncertainty as physically meaningful.Source: §6.1 · p. 9
6.2 Assessing Uncertainty Estimates:
- In Figure 8, ground-truth coverage within sample percentiles is plotted against percentile; baselines lie slightly below the ideal diagonal, while PGA-LSTM lies slightly above it.Thus, baseline uncertainty is slightly over-confident and PGA-LSTM uncertainty slightly under-confident; PGA-LSTM nevertheless has lower Per Sample Test RMSE than the baselines.Source: §6.2 · pp. 9–10, Figure 8 · p. 10
7 Conclusions and Future Work
The paper develops a physics-guided LSTM for lake temperature modeling, embedding monotonicity-preserving structure to predict physically consistent densities. The authors report generalizable, physically consistent solutions under MC dropout and identify broader applications and physical relationships as future work.
Conclusions
- The PGA-LSTM uses a monotonicity-preserving LSTM module to predict physically consistent densities for lake temperature modeling.Source: §7 · p. 10
- Comparisons with baselines demonstrate generalizable and physically consistent solutions despite minor weight perturbations from MC dropout.Source: §7 · p. 10
Future Work
- Future work will apply PGA-LSTM to other problems with monotonic recurrence relationships and explore other uncertainty methods, more complex space-time physics, and physical interpretability.Source: §7 · p. 10