Are Statistical Methods Obsolete in the Era of Deep Learning? A Study of ODE Inverse Problems

arXiv stat.ML / 4/6/2026

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Key Points

The study compares deep learning via a physics-informed neural network (PINN) against statistically principled inference using manifold-constrained Gaussian process inference (MAGI) on mechanistic nonlinear ODE inverse problems.
Using the SEIR epidemiology model and the Lorenz chaotic dynamics model, the authors find that statistical methods remain effective—particularly under sparse and noisy observations.
For parameter inference and trajectory reconstruction, MAGI achieves lower bias and variance than PINNs while using fewer parameters and requiring less hyperparameter tuning.
The results indicate statistical methods can outperform deep learning models for out-of-sample future prediction, where limited data causes overparameterized deep models to generalize poorly.
The paper also reports that statistically principled approaches are more robust to numerical imprecision and better align with the true governing ODE structure.

Abstract

In the era of AI, neural networks have become increasingly popular for modeling, inference, and prediction, largely due to their potential for universal approximation. With the proliferation of such deep learning models, a question arises: are leaner statistical methods still relevant? To shed insight on this question, we employ the mechanistic nonlinear ordinary differential equation (ODE) inverse problem as a testbed, using the physics-informed neural network (PINN) as a representative of the deep learning paradigm and manifold-constrained Gaussian process inference (MAGI) as a representative of statistically principled methods. Through case studies involving the SEIR model from epidemiology and the Lorenz model from chaotic dynamics, we demonstrate that statistical methods are far from obsolete, especially when working with sparse and noisy observations. On tasks such as parameter inference and trajectory reconstruction, statistically principled methods consistently achieve lower bias and variance, while using far fewer parameters and requiring less hyperparameter tuning. Statistical methods can also decisively outperform deep learning models on out-of-sample future prediction, where the absence of relevant data often leads overparameterized models astray. Additionally, we find that statistically principled approaches are more robust to accumulation of numerical imprecision and can represent the underlying system more faithfully to the true governing ODEs.