ArticleProceedings of the National Academy of Sciences of the United States of America2022
The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale's 18th problem.
Article in Proceedings of the National Academy of Sciences of the United States of America, 2022. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 21 papers.
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21 citing papers in PubMed.
- Transcription clusters and developmental pathways - nature, nurture, noise.Journal of cell science · 2026Review
- Corrosion fatigue life prediction of multiple austenitic stainless steel grades in high-temperature, high-pressure water environments using a GA-optimized neural network.Scientific reports · 2025Article
- Soft Bioelectronic Interfaces for Continuous Peripheral Neural Signal Recording and Robust Cross-Subject Decoding.Advanced science (Weinheim, Baden-Wurttemberg, Germany) · 2025Article
- On the uncertainty principle of neural networks.iScience · 2025Article
- The Detection of COVID-19-Related Multivariate Biomarker Immune Response in Pediatric Patients: Statistical Aspects.Viruses · 2025Article
- Data Flow-Based Strategies to Improve the Interpretation and Understanding of Machine Learning Models.Bioengineering (Basel, Switzerland) · 2024Article
- Exploring the uncertainty principle in neural networks through binary classification.Scientific reports · 2024Article
- Quantum-inspired analysis of neural network vulnerabilities: the role of conjugate variables in system attacks.National science review · 2024Article
- Singular-Value-Decomposition-Based Matrix Surgery.Entropy (Basel, Switzerland) · 2024Article
- The limit of human intelligence.Heliyon · 2024Article
- Clinical Super-Resolution Computed Tomography of Bone Microstructure: Application in Musculoskeletal and Dental Imaging.Annals of biomedical engineering · 2024Article
- Integrating Artificial Intelligence Tools in the Clinical Research Setting: The Ovarian Cancer Use Case.Diagnostics (Basel, Switzerland) · 2023Article
- Physics-supervised deep learning-based optimization (PSDLO) with accuracy and efficiency.Proceedings of the National Academy of Sciences of the United States of America · 2023Article
- A convolutional neural network provides a generalizable model of natural sound coding by neural populations in auditory cortex.PLoS computational biology · 2023Article
- Mapping the human connectome using diffusion MRI at 300 mT/m gradient strength: Methodological advances and scientific impact.NeuroImage · 2022Review
- Stabilizing deep tomographic reconstruction: Part A. Hybrid framework and experimental results.Patterns (New York, N.Y.) · 2022Article
- Stabilizing deep tomographic reconstruction: Part B. Convergence analysis and adversarial attacks.Patterns (New York, N.Y.) · 2022Article
- The Free Energy Principle for Perception and Action: A Deep Learning Perspective.Entropy (Basel, Switzerland) · 2022Review
- A phase transition for finding needles in nonlinear haystacks with LASSO artificial neural networks.Statistics and computing · 2022Article
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3 authors.
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Abstract
Deep learning (DL) has had unprecedented success and is now entering scientific computing with full force. However, current DL methods typically suffer from instability, even when universal approximation properties guarantee the existence of stable neural networks (NNs). We address this paradox by demonstrating basic well-conditioned problems in scientific computing where one can prove the existence of NNs with great approximation qualities; however, there does not exist any algorithm, even randomized, that can train (or compute) such a NN. For any positive integers K>2 and L, there are cases where simultaneously 1) no randomized training algorithm can compute a NN correct to K digits with probability greater than 1/2; 2) there exists a deterministic training algorithm that computes a NN with K –1 correct digits, but any such (even randomized) algorithm needs arbitrarily many training data; and 3) there exists a deterministic training algorithm that computes a NN with K –2 correct digits using no more than L training samples. These results imply a classification theory describing conditions under which (stable) NNs with a given accuracy can be computed by an algorithm. We begin this theory by establishing sufficient conditions for the existence of algorithms that compute stable NNs in inverse problems. We introduce fast iterative restarted networks (FIRENETs), which we both prove and numerically verify are stable. Moreover, we prove that only O(|log (ϵ)|) layers are needed for an ϵ-accurate solution to the inverse problem.
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