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Deep Neural Networks as Discrete Dynamical Systems: Implications for Physics-Informed Learning

Abhisek Ganguly, Santosh Ansumali, Sauro Succi · Jan 1, 2026 · Citations: 0

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What to verify

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Evidence quality

Provisional

Derived from abstract and metadata only.

Abstract

We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms. A comparative analysis between the numerical/exact solutions of the Burgers' and Eikonal equations, and the same obtained via PINNs is presented. We show that PINN learning provides a different computational pathway compared to standard numerical discretization in approximating essentially the same underlying dynamics of the system. Within this framework, DNNs can be interpreted as discrete dynamical systems whose layer-wise evolution approaches attractors, and multiple parameter configurations may yield comparable solutions, reflecting the non-uniqueness of the inverse mapping. In contrast to the structured operators associated with finite-difference (FD) procedures, PINNs learn dense parameter representations that are not directly associated with classical discretization stencils. This distributed representation generally involves a larger number of parameters, leading to reduced interpretability and increased computational cost. However, the additional flexibility of such representations may offer advantages in high-dimensional settings where classical grid-based methods become impractical.

Abstract-only analysis — low confidence

All signals on this page are inferred from the abstract only and may be inaccurate. Do not use this page as a primary protocol reference.

  • This page is still relying on abstract and metadata signals, not a fuller protocol read.
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Should You Rely On This Paper?

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Best use

Background context only

Use if you need

A provisional background reference while structured extraction finishes.

Main weakness

This page is still relying on abstract and metadata signals, not a fuller protocol read.

Trust level

Provisional

Usefulness score

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Eval-fit score is unavailable until extraction completes.

Human Feedback Signal

Not explicit in abstract metadata

Evaluation Signal

Weak / implicit signal

Usefulness for eval research

Provisional (processing)

Extraction confidence 0%

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What We Could Verify

These are the protocol signals we could actually recover from the available paper metadata. Use them to decide whether this paper is worth deeper reading.

Human Feedback Types

provisional (inferred)

None explicit

No explicit feedback protocol extracted.

"We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms."

Evaluation Modes

provisional (inferred)

None explicit

Validate eval design from full paper text.

"We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms."

Quality Controls

provisional (inferred)

Not reported

No explicit QC controls found.

"We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms."

Benchmarks / Datasets

provisional (inferred)

Not extracted

No benchmark anchors detected.

"We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms."

Reported Metrics

provisional (inferred)

Not extracted

No metric anchors detected.

"We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms."

Rater Population

provisional (inferred)

Unknown

Rater source not explicitly reported.

"We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms."

Human Feedback Details

This page is using abstract-level cues only right now. Treat the signals below as provisional.

  • Potential human-data signal: No explicit human-data keywords detected.
  • Potential benchmark anchors: No benchmark names detected in abstract.
  • Abstract highlights: 3 key sentence(s) extracted below.

Evaluation Details

Evaluation fields are inferred from the abstract only.

  • Potential evaluation modes: No explicit eval keywords detected.
  • Potential metric signals: No metric keywords detected.
  • Confidence: Provisional (metadata-only fallback).

Research Brief

Metadata summary

We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms.

Based on abstract + metadata only. Check the source paper before making high-confidence protocol decisions.

Key Takeaways

  • We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms.
  • A comparative analysis between the numerical/exact solutions of the Burgers' and Eikonal equations, and the same obtained via PINNs is presented.
  • We show that PINN learning provides a different computational pathway compared to standard numerical discretization in approximating essentially the same underlying dynamics of the system.

Researcher Actions

  • Compare this paper against nearby papers in the same arXiv category before using it for protocol decisions.
  • Check the full text for explicit evaluation design choices (raters, protocol, and metrics).
  • Use related-paper links to find stronger protocol-specific references.

Caveats

  • Generated from abstract + metadata only; no PDF parsing.
  • Signals below are heuristic and may miss details reported outside the abstract.

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