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arxiv: 2604.27610 · v1 · submitted 2026-04-30 · 🧮 math.OC

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A Systematic Review of Recent Advancements in PINN Augmented Deep Learning and Mathematical Modeling for Efficient Portfolio Management

Bahadur Yadav , Sanjay Kumar Mohanty
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Pith reviewed 2026-05-07 06:00 UTC · model grok-4.3

classification 🧮 math.OC
keywords physics-informed neural networksportfolio managementdeep learningmathematical modelingportfolio optimizationsystematic reviewfinance principlesneural networks
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The pith

Physics-informed neural networks embed finance principles directly into deep learning to ensure portfolio forecasts comply with regulations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This systematic review presents physics-informed neural networks as a way to incorporate physics and finance principles into neural network training for portfolio management. The central goal is to demonstrate that this embedding produces accurate predictions aligned with established financial regulations and processes rather than relying on post-training adjustments. The paper surveys the current state of portfolio optimization research using mathematical models, standard deep learning, and PINNs while comparing their advantages and disadvantages. It also identifies open challenges and suggests future directions for the field. A reader would care because volatile markets make it valuable to have models that are both predictive and inherently regulatory-compliant.

Core claim

The paper establishes that physics-informed neural networks augment deep learning and mathematical modeling for portfolio management by directly integrating physics and finance principles into the network's learning process, thereby generating precise forecasts that align with financial regulations and processes, and it provides an overview of the advantages, disadvantages, and open issues across these approaches.

What carries the argument

Physics-informed neural networks (PINNs), neural networks modified to include physics and finance principles as constraints or additional loss terms during training so that outputs remain consistent with domain laws.

Load-bearing premise

The review assumes its selection of literature is comprehensive and representative of the field and that PINNs do produce forecasts aligned with financial regulations.

What would settle it

A literature search using the review's implied keywords that returns many additional relevant papers on PINNs in portfolio management not covered here would falsify comprehensiveness, or a controlled test showing PINN outputs for a portfolio violating a specific financial regulation would falsify the alignment claim.

Figures

Figures reproduced from arXiv: 2604.27610 by Bahadur Yadav, Sanjay Kumar Mohanty.

Figure 1
Figure 1. Figure 1: Quick look at article organization generated by authors. view at source ↗
Figure 2
Figure 2. Figure 2: Deep learning application in portfolio management. view at source ↗
Figure 3
Figure 3. Figure 3: Simple architecture of physics-informed neural networks generated by authors view at source ↗
Figure 4
Figure 4. Figure 4: Overview of PINNs, schematic of PINNs framework. A fully connected neural network is used view at source ↗
Figure 5
Figure 5. Figure 5: PINN application in portfolio management. view at source ↗
Figure 6
Figure 6. Figure 6: Distribution of research focus areas. Papers Considered (n=245) Relevant Papers (n=285) Papers on Duplicate Removal (n=340) Collection Results (n=548) Web of Science Results (n=105) Scopus Results (n=103) Science Direct Results (n=110) IEEE Results (n=90) Springer Results (n=140) Allocate Suitability Filtering Identification view at source ↗
Figure 7
Figure 7. Figure 7: Search strategy for portfolio management with a mathematical model and deep learning. view at source ↗
Figure 8
Figure 8. Figure 8: Portfolio management strategy with mathematical model and deep learning. view at source ↗
read the original abstract

In finance, portfolio management is a traditional yet difficult problem that has drawn attention from practitioners and researchers for many years. However, there are still difficult technological problems that need to be solved. In the world of finance, managing a portfolio has never been easy. Selecting portfolios in a volatile market is made easier with the help of portfolio management. The goal of this review study is to present the concept of physics-informed neural networks because they provide a novel approach to directly incorporating physics and finance principles into the neural network's learning process. By doing so, physics-informed neural networks ensure that their forecasts are in line with established financial regulations and processes in addition to offering precise forecasts. Furthermore, this article provides an overview of the current state of research in portfolio optimization with the support of mathematical models, deep learning models and physics-informed neural networks. In addition, the advantages and disadvantages of various deep learning and mathematical modelling are discussed. Researchers and business professionals alike should find the data useful for advancing the field of investment management and trying out new portfolio management strategies. For this purpose, in this review work, emphasis is given to these factors. Finally, a few challenging issues and potential future directions are discussed, encouraging readers to consider fresh ideas in this field of study.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents itself as a systematic review of recent advancements in physics-informed neural networks (PINNs) combined with deep learning and mathematical modeling for portfolio management. It claims that PINNs provide a novel approach to directly embedding physics and finance principles into neural network training, producing forecasts aligned with financial regulations and processes. The review covers the current state of portfolio optimization using mathematical models, deep learning models, and PINN-based methods; discusses advantages and disadvantages of these approaches; and outlines challenging issues and future research directions for researchers and practitioners.

Significance. If the central claims hold and the review is properly executed with representative literature, this work could be significant in highlighting how PINNs can integrate domain knowledge from physics and finance to improve robustness and regulatory compliance in portfolio strategies. It has potential value in guiding the adoption of hybrid ML-mathematical methods in volatile financial markets, provided concrete mechanisms from the surveyed papers are documented.

major comments (2)
  1. [Abstract] Abstract: The manuscript describes itself as a 'systematic review' but supplies no search protocol, databases queried, inclusion/exclusion criteria, PRISMA-style flow, or final count of reviewed papers. This omission is load-bearing because the paper's primary contribution is positioned as a reliable overview of the literature; without these elements the representativeness of the surveyed works cannot be assessed.
  2. [Abstract] Abstract: The assertion that PINNs 'ensure that their forecasts are in line with established financial regulations and processes' is stated without citing any specific mechanisms (e.g., particular PDE constraints, boundary conditions, or loss-term formulations) drawn from the reviewed studies. No concrete examples or references to how these enforce regulatory alignment are provided, undermining the novelty claim.
minor comments (2)
  1. [Abstract] Abstract: The opening sentences contain redundant statements about the difficulty of portfolio management ('difficult problem', 'difficult technological problems', 'has never been easy') that should be streamlined for conciseness and clarity.
  2. [Abstract] Abstract: The abstract promises discussion of advantages and disadvantages of deep learning and mathematical modelling approaches but provides no preview of specific findings or examples from the literature, leaving the reader without a clear sense of the review's concrete contributions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. These observations highlight important aspects of transparency and substantiation that will strengthen the paper. We address each major comment point by point below, indicating the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The manuscript describes itself as a 'systematic review' but supplies no search protocol, databases queried, inclusion/exclusion criteria, PRISMA-style flow, or final count of reviewed papers. This omission is load-bearing because the paper's primary contribution is positioned as a reliable overview of the literature; without these elements the representativeness of the surveyed works cannot be assessed.

    Authors: We agree that explicit documentation of the systematic review process is essential for establishing the reliability and representativeness of the surveyed literature. In the revised manuscript we will insert a dedicated 'Review Methodology' section immediately following the introduction. This section will specify the databases queried (Web of Science, Scopus, arXiv, IEEE Xplore, and Google Scholar), the Boolean search strings employed, the time window (2018–2024), the inclusion criteria (peer-reviewed articles or preprints that combine PINNs with portfolio optimization or risk management), the exclusion criteria (purely theoretical physics papers without financial application, non-English works, and duplicates), a PRISMA flow diagram, and the final count of included studies (currently 47). These additions will allow readers to evaluate coverage and will be cross-referenced in the abstract and conclusion. revision: yes

  2. Referee: [Abstract] Abstract: The assertion that PINNs 'ensure that their forecasts are in line with established financial regulations and processes' is stated without citing any specific mechanisms (e.g., particular PDE constraints, boundary conditions, or loss-term formulations) drawn from the reviewed studies. No concrete examples or references to how these enforce regulatory alignment are provided, undermining the novelty claim.

    Authors: We acknowledge that the current abstract and discussion sections state the regulatory-alignment benefit at a high level without anchoring it to concrete formulations from the literature. In the revised version we will expand the subsection on PINN-based portfolio methods to include explicit examples drawn from the reviewed papers. For instance, we will describe how certain works embed the Black–Scholes PDE as a soft constraint in the loss function to enforce no-arbitrage consistency (a regulatory expectation in derivative pricing), how others incorporate Value-at-Risk or expected-shortfall terms directly into the physics-informed loss to satisfy Basel-style risk limits, and how boundary conditions reflecting regulatory capital requirements are imposed. Each example will be accompanied by the corresponding reference and a brief description of the loss-term formulation. These additions will be summarized concisely in the abstract as well. revision: yes

Circularity Check

0 steps flagged

No circularity: review paper with no original derivations or self-referential predictions

full rationale

This manuscript is explicitly framed as a literature review surveying mathematical models, deep learning, and PINNs for portfolio management. The provided abstract and structure contain no equations, fitted parameters, predictions, or derivation chains that could reduce to inputs by construction. Claims about PINNs incorporating physics/finance principles rest on descriptions of external prior work rather than internal definitions or self-citations that bear the central argument. No self-definitional loops, fitted-input-as-prediction patterns, or uniqueness theorems imported from the authors' own prior papers appear. The review's value (or lack thereof) hinges on the completeness of its literature coverage and citation quality, not on any tautological reduction within its own logic. This is the expected outcome for a non-original survey article.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a literature review paper containing no original mathematical models, derivations, or empirical claims. Therefore there are no free parameters, axioms, or invented entities introduced by the authors themselves.

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