What You See is Not What You Get: Neural Partial Differential Equations and The Illusion of Learning

Abstract: Differentiable Programming for scientific machine learning (SciML) has recently seen considerable interest and success, as it directly embeds neural networks inside PDEs, often called as NeuralPDEs, derived from first principle physics. Therefore, there is a widespread assumption in the community that NeuralPDEs are more trustworthy and generalizable than black box models. However, like any SciML model, differentiable programming relies predominantly on high-quality PDE simulations as "ground truth" for training. However, mathematics dictates that these are only discrete numerical approximations of the true physics. Therefore, we ask: Are NeuralPDEs and differentiable programming models trained on PDE simulations as physically interpretable as we think? In this work, we rigorously attempt to answer these questions, using established ideas from numerical analysis, experiments, and analysis of model Jacobians. Our study shows that NeuralPDEs learn the artifacts in the simulation training data arising from the discretized Taylor Series truncation error of the spatial derivatives. Additionally, NeuralPDE models are systematically biased, and their generalization capability is likely enabled by a fortuitous interplay of numerical dissipation and truncation error in the training dataset and NeuralPDE, which seldom happens in practical applications. This bias manifests aggressively even in relatively accessible 1-D equations, raising concerns about the veracity of differentiable programming on complex, high-dimensional, real-world PDEs, and in dataset integrity of foundation models. Further, we observe that the initial condition constrains the truncation error in initial-value problems in PDEs, thereby exerting limitations to extrapolation. Finally, we demonstrate that an eigenanalysis of model weights can indicate a priori if the model will be inaccurate for out-of-distribution testing.

• The paper reveals that NeuralPDEs often learn discretization artifacts instead of underlying physical phenomena, challenging assumptions about model generalization.
• It demonstrates that initial conditions heavily influence truncation errors, reducing the reliability of extrapolations in practical scenarios.
• An eigenvalue analysis shows that numerical biases from training simulations can compromise model stability and predictive performance.

An Evaluation of Neural Partial Differential Equations: Numerical Bias and Generalization

The research paper, "What You See is Not What You Get: Neural Partial Differential Equations and The Illusion of Learning," presents a comprehensive examination of Neural Partial Differential Equations (NeuralPDEs) within scientific machine learning frameworks. This investigation focuses on the intrinsic numerical errors arising from the discretization processes typically employed in computational simulations and how these errors manifest in the learning dynamics and generalization capability of NeuralPDEs as applied to partial differential equations (PDEs).

Core Investigative Approach and Methodology

The study employs both the viscous Burgers equation and the geophysical Korteweg de Vries (gKdV) equation as test cases. These one-dimensional PDEs are chosen for their simplicity and ability to exhibit a range of behaviors due to varying levels of viscous and dispersive properties. The authors question the default assumption that NeuralPDEs, by virtue of embedding known physics within their framework, provide inherently interpretable and generalizable results. They explore this assumption through rigorous mathematical analysis and numerical experimentation.

A critical component of the methodology is the recognition that training datasets used in scientific machine learning often rely on numerical simulations where PDE components are represented by truncated Taylor Series expansions. These approximations inherently introduce discretization errors that NeuralPDE models learn alongside the physics-based solutions. The research addresses whether these models are genuinely learning the underlying physics or primarily capturing numerical artifacts that may lead to misleading inferences regarding NeuralPDE generalization and interpolation.

Key Findings

1. Learning the Wrong Features: The study finds that NeuralPDEs are prone to learning discretization artifacts rather than purely representing physical phenomena. When extrapolating to unseen conditions, the models often exhibit biased generalization as a result of these artifacts.
2. Impact of Initial Conditions: The initial conditions play a crucial role in influencing the truncation error, thereby affecting the model’s capability to extrapolate. This finding challenges the perceived robustness of NeuralPDEs when the training initial conditions differ significantly from the testing scenarios.
3. Numerical Bias and Truncation Error Sensitivity: Numerical schemes used for derivatives in the training simulations introduce biases which can substantially alter the expected performance of NeuralPDEs. Mismatches in the numerical treatment between training data and model lead to erroneous learning and prediction cycles.
4. Eigenvalue Analysis for Predicting Model Stability: An eigenanalysis of learned network weights reveals insights into model stability and provides an indicator of generalization potential and limitations before extensive out-of-distribution testing is conducted.

Implications and Speculative Outlook

This research posits a significant paradigm shift in evaluating the integrity and utility of NeuralPDEs. It forewarns of the dangers associated with assuming the fidelity of data derived from various numerical simulations as ground truth. The implications are critical for the advancement of foundation models in scientific applications, where datasets comprised of simulations from diverse numerical setups could introduce hidden biases.

Further research could expand upon this work by exploring adaptive learning strategies that mitigate learning numerical artifacts while strengthening the model’s focus on physically relevant features. Practical advancements may see the integration of meta-learning or mixed-integer optimization methods to dynamically adjust network parameters based on real-time error analysis.

In conclusion, while NeuralPDEs offer immense promise in scenario analysis and rapid forecasting across sciences, their reliability hinges on recognizing and correcting the biases introduced by truncation errors and initial conditions. Future methodologies would benefit from incorporating diagnostic tools for identifying potential pitfalls in the training phase, ensuring NeuralPDEs evolve toward more robust applications in modeling complex dynamical systems.