Pith. sign in

REVIEW 4 major objections 4 minor 33 references

Perception-Informed Neural Networks: Beyond Physics-Informed Neural Networks

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper's thesis is that perception-informed neural networks (PrINNs) unify PINNs and their variants: any system description—crisp equation, probability, possibility, interval, fuzzy graph, or mixture—can be precisiated into a…

desk verdict A useful taxonomy of informed neural networks, but the sureness loss is mathematically flawed and the paper overclaims with zero experiments. read the letter →

arxiv 2505.03806 v2 pith:CCOR5OMK submitted 2025-05-02 cs.LG cs.AIcs.NE

Mehran Mazandarani , Marzieh Najariyan This is my paper · ORCID
classification cs.LGcs.AIcs.NE
keywords ArtificialIntelligenceMachineLearningComputationalScienceandEngineeringGeneralizedTheoryofUncertaintyDeepNeuralNetworksPerceptionComputingDynamicalSystemsPhysics-Informed
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Perception-Informed Neural Networks (PrINNs) are proposed as a general recipe for telling a neural network what a human knows about a dynamical system, even when that knowledge is imprecise and cannot be written as a crisp differential equation. The recipe is to express the knowledge as a natural-language proposition, precisiate it into a generalized constraint of the form $X \text{ is}_r R$ with a chosen mode, and add a loss term that penalizes violation of that constraint. The paper argues that standard physics-informed neural networks are exactly this construction in the singular mode, so every existing PINN variant becomes a special case rather than a separate idea. It also introduces named variants: mixture-of-experts informed neural networks, transformed-knowledge informed neural networks, and fuzzy-informed neural networks, which turn fuzzy if-then rules into differentiable penalties without defuzzification. If the recipe works, it gives a single design space for data-driven models that can use words, expert opinions, and uncertain measurements alongside equations.

What carries the argument

The load-bearing object is the generalized constraint, written as $GC(p): X \text{ is}_r R$, where $X$ is the state or parameter being described, $r$ is the mode of precisiation, and $R$ is the constraining relation such as a normal distribution, a triangular fuzzy number, or a fuzzy graph. The argument runs on three concrete loss-building devices: the horizontal membership function of a fuzzy residual, whose granules are crisp functions of membership degree and a relative-distance-measure variable; the possibility factor $M^{1-\mu}$, which down-weights granules far from the most plausible value; and the fuzzy-graph penalty $M(1-R)^2$, which penalizes the network whenever the joint membership of input and output in the rule base is less than one. For sureness-informed networks, the mechanism is the scalar $\text{sureness} = \mu \times N_{g_i}$, the product of the possibility degree and the probability-density value of the residual, which encodes the principle that impossibility implies improbability. These objects do the work of converting words and imprecise statements into gradients.

What would settle it

Train a PrINN on a system whose true dynamics are known, with a fuzzy if-then rule that contradicts a plausible data-driven fit, such as “if $t$ is small then $x(t)$ is large” while the data say $x(t)$ is small for small $t$. If the $M(1-R)^2$ fuzzy-graph penalty does not measurably change the learned trajectory toward the rule as $M$ grows, the central assumption fails. A sharper check: initialize the sureness loss with $\mu = 0$ or $N_{g_i} = 0$; since sureness is the product, the gradient of $(1-\text{sureness})^2$ vanishes at those points, so no learning step can increase sureness, and training should stall exactly there.

Watch

Extended reading notes

Core claim

The paper's central claim is that “informed” neural networks, those whose loss includes knowledge about the system in addition to data, are all instances of one construction. Start from an imprecise differential equation, in which parameters, initial conditions, or whole dynamics are described by propositions such as “the damping ratio is approximately 0.2”; precisiate that description by choosing a mode of the generalized theory of uncertainty; then build the loss from the precisiated residuals. The resulting network is a PrINN. The paper further argues that non-homogeneous precisiation, with different parameters in different modes, yields new kinds of differential equations such as Z-differential equations and fuzzy-probabilistic differential equations, and that these can be solved or discovered by appropriate PrINNs. It also maintains that perception-based rules can replace physics laws entirely when no equation is known, with fuzzy if-then rules encoded directly as loss penalties.

Load-bearing premise

The load-bearing premise is that a perception-based proposition, once precisiated into a generalized constraint, can be turned into a differentiable loss penalty that gradient descent reliably satisfies, and that the proposed penalty forms $M^{1-\mu}$, $M(1-R)^2$, and sureness $\mu \times N_{g_i}$ train stably without special measures; the paper provides no convergence analysis, experiments, or robustness discussion for this premise.

Editorial extensions

If this is right

  • If PrINNs are correct, PINNs, physics-informed Kolmogorov-Arnold networks, fuzzy PINNs, Bayesian PINNs, and similar variants are all recoverable by fixing the precisiation mode, so new informed networks can be designed by choosing a mode rather than by inventing a new architecture.
  • The fuzzy-graph mode gives a way to train a controller or model online from expert rules without defuzzification or pre-training, because the rules enter only through a differentiable penalty.
  • Mixture-of-experts informed neural networks allow parameters in the same differential equation to be precisiated in different modes, which the paper argues can represent and discover differential equations that have not been studied before, such as Z-differential equations and fuzzy-probabilistic differential equations.
  • Transformed-knowledge informed neural networks can constrain predictions with meta-information such as sureness, sureness time, or compatibility of precisiations, so the network not only satisfies equations but stays within a consensus region of expert views.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The decisive next step, which the paper does not take, is a benchmark where fuzzy-rule constraints are varied while data are held fixed; without such a test, the PrINN construction remains a taxonomy rather than a demonstrated learning method.
  • The sureness product $\mu \times N_{g_i}$ suggests an engineering caution: if either factor is near zero, the gradient of $(1-\text{sureness})^2$ nearly vanishes, so practical implementations would need a floor, rescaling, or a different consistency measure.
  • The same construction could absorb many non-physics soft constraints, such as monotonicity, safety envelopes, or expert heuristics, by phrasing them as propositions and choosing a precisiation mode, making PrINNs a general constraint-injection language for deep learning.
  • A testable consequence of the mixture-of-modes claim is that solving a Z-differential equation with a mixture-of-experts informed network should reproduce both the probabilistic envelope and the possibilistic envelope of the solution; obtaining only one envelope would indicate the mixture loss is not enforcing both constraints.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper introduces Perception-Informed Neural Networks (PrINNs), a conceptual framework intended to generalize Physics-Informed Neural Networks (PINNs) by incorporating perception-based information into neural network training through loss functions. The framework is organized around the generalized theory of uncertainty (GTU): a perception expressed as a proposition is precisiated into a generalized constraint (singular, probability, possibility, interval, fuzzy graph, etc.), and the corresponding constraint is added to the total loss. The manuscript presents several instantiations: PINNs as the singular mode, fuzzy calculus-informed neural networks (FcINNs) for possibility distributions, and mixture-of-experts informed neural networks (MOEINNs) that combine heterogeneous precisiations. It also introduces transformed-knowledge informed neural networks (TKINNs) and sureness-informed neural networks (SINNets) based on the product of possibility and probability, and fuzzy-informed neural networks (FINNs) that encode fuzzy if-then rules as penalty terms, with a claimed application to online data-driven control without defuzzification. No simulations, datasets, convergence analyses, or comparisons are provided.

Significance. If the proposed loss formulations were sound, the paper would make a useful contribution by providing a unified language and taxonomy for informed neural networks, bridging fuzzy logic, the generalized theory of uncertainty, and deep learning. The FINN idea of embedding fuzzy rules directly into the loss without a defuzzification module is conceptually appealing and could have practical value for fuzzy control. However, the current version's central capabilities are asserted rather than demonstrated, and at least one flagship loss (the sureness loss) is mathematically flawed. The framework also relies on several ad-hoc design choices (e.g., the possibility factor M^{1-µ} and the penalty weight M) whose behavior is not analyzed. Thus, while the conceptual organization has merit, the technical foundations are not yet established, and the paper would need substantial revision to support its claims.

major comments (4)
  1. [§3.1, Eqs. (27), (32), (33)] The sureness loss L_s = (1 - sureness)^2 is ill-posed because N_gi in Eq. (32) is a probability density, not a probability, and is not bounded above by 1. For the manuscript's own σ = 0.01 example, the density's maximum is 1/(√(2π)σ) ≈ 39.89 when |2ω_n ẋ| = 1, so sureness = μ N_gi can exceed 1 for every μ ∈ [0,1]. At the exact solution g = 0, the loss is minimized by μ = 1/N_gi rather than μ = 1, so the learnable possibility degree is pushed away from the expert's modal value; conversely, if μ is fixed, the residual is driven to the contour N_gi = 1/μ, which lies away from the modal residual. The statement that N_g maps to [0,1] is therefore false, and the loss does not implement the 'impossibility implies improbability' principle. The authors should redefine sureness using a probability (e.g., the cumulative probability over an interval around the residual) or otherwise normalize the density.
  2. [§3.2, Eq. (34) and Eqs. (35), (37), (39)] The FINN loss L_R = M(1 - R)^2 presupposes that the overall restriction R lies in [0,1], but the paper's examples define R as an algebraic sum of t-norm values (e.g., Eq. (35): R = μ_small(t)∧μ_large(x) + μ_large(t)∧μ_medium(x)). With two or more rules firing simultaneously, this sum can exceed 1, so the loss would be minimized by reducing R to 1 rather than by maximizing rule satisfaction. The disjunction should be a proper s-norm such as maximum (as in the control example of Fig. 6) or the range of R should be established and the loss adjusted accordingly.
  3. [Abstract and §3.2] The paper asserts several concrete capabilities that are not demonstrated: FINNs 'allowing online training without pre-training' and 'can be applied directly online' as a real-time controller, and MOEINNs enabling 'discovery of previously unknown forms of differential equations.' No experiments, datasets, comparisons, or convergence analyses are provided for any of the proposed loss functions. Since the central claim is that perception constraints can be turned into differentiable penalties that gradient descent will reliably satisfy, at least a proof-of-concept experiment or a formal convergence/stability analysis is required to make the claim credible. If the paper is intended as a purely conceptual position paper, these assertions should be qualified as possibilities rather than demonstrated capabilities.
  4. [§3.1, Eq. (24)] The possibility factor M^{1-μ} in the FcINN loss weights low-membership granules (μ < 1) more heavily than the modal granule, but the accompanying text says that the degree of perception satisfaction decreases as μ decreases. If the intent is to relax the constraint for less possible values, the factor should be nondecreasing in μ (e.g., μ or μ^2); as written, the network is forced to satisfy the differential equation more strongly exactly for the values that are least consistent with the perception. The authors should either correct the weighting or provide a clear interpretation of why low-possibility granules should receive larger penalties.
minor comments (4)
  1. [§2.1, Eq. (6)] In the triangular fuzzy number for the damping ratio, ζ = (0.08, 0.1, 1.01), the right endpoint 1.01 is likely a typo; it should presumably be 0.12. Please check.
  2. [§2.2, Eq. (26)] The notation 'p1/less than 5' is not formally defined. If p_i denotes the probability of a fuzzy event (e.g., 'less than 5'), the authors should specify how probabilities are assigned to fuzzy sets, since standard probability theory does not attach probabilities to fuzzy propositions without a definition (e.g., via the Zadeh probability of a fuzzy event).
  3. [§2.1, Eq. (7)] The symbol 'Z+-number' is used without definition; the surrounding text defines Z-numbers (Ã, p_x). Please clarify whether Z+ is a distinct concept or a typo for Z-number.
  4. [References] Reference formatting is inconsistent; for example, [8] lacks full venue details, and [23] uses a nonstandard journal name. Please standardize according to the journal's style.

Circularity Check

1 steps flagged · score 4.0 of 10

The central capability claim is definitional: PrINNs are defined as networks that integrate perception into the loss, and the ability to model by integrating perception into the loss is then restated as the framework's contribution.

  1. self definitional [Abstract; Section 3 (definition of PrINNs, page 15)]
    "PrINNs allow neural networks to model dynamical systems by integrating expert knowledge and perception-based information through loss functions, enabling the creation of modern data-driven models."

    Section 3 defines PrINNs as neural networks informed by perception through words or propositions and states that the general approach to setting up PrINNs is to integrate the constraint R into the loss function. The abstract's sentence that PrINNs allow modeling by integrating perception-based information through loss functions is thus a restatement of this definition rather than a derived result. The new variants (MOEINNs, TKINNs, FINNs) are introduced by writing loss terms such as Ls = (1 - sureness)^2 and LR = M(1-R)^2, which contain the target constraint by construction. With no experiments, convergence analysis, or external benchmark supplied, the asserted modeling capability is contained in the definitions, not derived from first principles.

full rationale

The paper is an explicit framework/position paper, so many contributions are definitions and loss proposals rather than tested predictions. The one genuine circularity is that the headline capability of PrINNs, modeling by integrating perception into the loss, is exactly how a PrINN is defined in Section 3, and the loss forms in Eqs. (24), (27), (30), and (34) are constructed to contain the constraints they are said to enforce. This gives the framework a self-definitional flavor. However, the paper also contains independent conceptual content: the precisiation taxonomy, the relation of PINN variants to modes of precisiation, and the explicit design of new loss terms, all of which could be validated or refuted by future empirical work. The self-citations to IDEs, ZDEs, and granular derivatives are used as background concepts rather than as a uniqueness or existence argument that forces the paper's choices, and Notes 3 and 4 explicitly limit the scope of the MOEINN and granular-derivative claims. The mathematical issue that N_gi in Eq. (33) is a probability density rather than a probability, so sureness is not bounded by 1 and L_s does not implement the impossibility-implies-improbability principle at the modal constraint, is a correctness concern rather than a circularity and does not increase the circularity score.

Assumptions & free parameters 3 free parameters · 5 assumptions · 5 invented entities

The paper's ledger is heavy on borrowed concepts from fuzzy mathematics and the authors' own prior works. No new parameters are fitted to external data, but the framework depends on unverified optimization assumptions and unspecified penalty and membership choices.

free parameters (3)
  • Penalty weight M in possibility-factor and fuzzy-graph losses = M > 1, not specified
    Introduced in Eqs. (24) and (34) to control how strongly fuzzy constraints are enforced; no value or tuning procedure is given, and the central behavior of the proposed losses depends on it.
  • Possibility factor exponent M^(1-µ) = M > 1, not specified
    Used in Eq. (24) and (30) to weight fuzzy residual losses by membership degree; the choice of M and the functional form f(1)=1 are ad hoc design decisions.
  • Fuzzy membership functions for illustrative rules = not specified
    The FINN examples require membership functions to compute R(t,x), but no membership functions, parameters, or learning procedures are provided.
assumptions (5)
  • domain assumption Zadeh's generalized theory of uncertainty and the generalized constraint form GC(p): X is_r R is a valid way to represent perception-based information.
    The whole PrINN framework is built on this precisiation process from [30,31]; if this representation is not appropriate, the framework lacks foundation.
  • domain assumption Horizontal membership functions (HMFs) and granular calculus provide correct crisp representations of fuzzy residuals.
    Equations (22)-(24) and (31) rely on HMF formalism from [18,19,10] to turn fuzzy differential equation residuals into functions of learnable µ and α.
  • ad hoc to paper A penalty of the form M(1-R)^2 enforces fuzzy if-then rule constraints under gradient descent.
    Eq. (34) is introduced as the FINN loss with no convergence or stability analysis; the claim that FINNs can be trained online without pre-training depends on this assumption.
  • domain assumption Sureness = possibility x probability is a meaningful meta-information quantity for training.
    Adopted from Zadeh [32] and used in Eq. (33) for SINNets; no empirical evidence is provided that this product improves learning.
  • domain assumption The min/max t-norm/s-norm used in fuzzy graphs is compatible with automatic differentiation and end-to-end training.
    FINN's R expression in Eqs. (35), (37), and (39) uses min/max, which are non-smooth; the paper assumes standard gradient-based training will still work.
invented entities (5)
  • PrINNs (Perception-Informed Neural Networks)
    purpose: Unifying framework for neural networks informed by perception-based information in various precisiation modes.
    Introduced as a new framework; no external benchmark or falsifiable prediction beyond unvalidated capabilities.
  • Imprecise differential equations (IDEs)
    purpose: Formal structure for differential equations with parameters and conditions described by propositions.
    Central conceptual foundation drawn from the authors' prior work [12]; no direct solution method exists, and the paper itself states there is no direct method.
  • MOEINNs
    purpose: Neural networks informed by a mixture of experts' modes of precisiation.
    Proposed as a model family; no experiments or comparisons.
  • TKINNs and SINNets
    purpose: Use meta-information such as sureness, defined as the product of possibility and probability, to improve predictions.
    Sureness is defined as µ x N_gi; no independent evidence that this quantity improves predictions.
  • FINNs
    purpose: Neural networks informed by fuzzy if-then rules through loss penalties, without a defuzzification step.
    Proposed as online trainable fuzzy deep networks; no controller simulation or real-world test.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Perception-Informed Neural Networks: Beyond Physics-Informed Neural Networks." pith.science (2026). https://pith.science/paper/CCOR5OMK

@misc{pith2026250503806,
  author       = {Pith},
  title        = {Pith review of: Perception-Informed Neural Networks: Beyond Physics-Informed Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CCOR5OMK}},
  note         = {Machine review of arXiv:2505.03806}
}
read the original abstract

This article introduces Perception-Informed Neural Networks (PrINNs), a framework designed to incorporate perception-based information into neural networks, addressing both systems with known and unknown physics laws or differential equations. Moreover, PrINNs extend the concept of Physics-Informed Neural Networks (PINNs) and their variants, offering a platform for the integration of diverse forms of perception precisiation, including singular, probability distribution, possibility distribution, interval, and fuzzy graph. In fact, PrINNs allow neural networks to model dynamical systems by integrating expert knowledge and perception-based information through loss functions, enabling the creation of modern data-driven models. Some of the key contributions include Mixture of Experts Informed Neural Networks (MOEINNs), which combine heterogeneous expert knowledge into the network, and Transformed-Knowledge Informed Neural Networks (TKINNs), which facilitate the incorporation of meta-information for enhanced model performance. Additionally, Fuzzy-Informed Neural Networks (FINNs) as a modern class of fuzzy deep neural networks leverage fuzzy logic constraints within a deep learning architecture, allowing online training without pre-training and eliminating the need for defuzzification. PrINNs represent a significant step forward in bridging the gap between traditional physics-based modeling and modern data-driven approaches, enabling neural networks to learn from both structured physics laws and flexible perception-based rules. This approach empowers neural networks to operate in uncertain environments, model complex systems, and discover new forms of differential equations, making PrINNs a powerful tool for advancing computational science and engineering.

Figures

Figures reproduced from arXiv: 2505.03806 by the authors.

Figure 1
Figure 1. Some of the modes of precisiation in generalized theory of uncertainty [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Some instances of IDEs. Thus, roughly speaking, IDEs are the result of integrating perception into differ￾ential equations, for which determining a solution requires a precise meaning of the propositions that convey perception. The process of precisiation of an IDE involves the transformation of perception-based information expressed about the differential equation entities. Therefore, any initial conditions, bounda… view at source ↗
Figure 3
Figure 3. The precisiation of IDEs in different modes. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Perception informed neural networks. The following introduces some PrINNs briefly, categorized based on the modes of precisiation and whether or not the system’s differential equations are known. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: FINN-based controller in a closed-loop control system. [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]
Figure 6
Figure 6. Figure 6: The overall restriction applied in the FINN-based controller. [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: The variants of Perception informed neural networks. [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 27 canonical work pages

  1. [1]

    Barros, F

    L.C. Barros, F. S. Pedro, Fuzzy differential equations with interactive derivative, Fuzzy Sets and Systems, 309, Pages 64-80, 2017

    work page 2017

  2. [2]

    S. Dong, N. Ni, A method for representing periodic functions and enforcing ex- actly periodic boundary conditions with deep neural networks, Journal of Com- putational Physics 435, 2021

    work page 2021

  3. [3]

    Fuhg, Ioannis Kalogeris, Am´ elie Fau, Nikolaos Bouklas, Interval and fuzzy physics-informed neural networks for uncertain fields, Probabilistic Engineering Mechanics, 68, 2022

    Jan N. Fuhg, Ioannis Kalogeris, Am´ elie Fau, Nikolaos Bouklas, Interval and fuzzy physics-informed neural networks for uncertain fields, Probabilistic Engineering Mechanics, 68, 2022

    work page 2022

  4. [4]

    Methods Appl

    Ling Guo, Hao Wu, Xiaochen Yu, Tao Zhou, Monte Carlo fPINNs: Deep learning method for forward and inverse problems involving high-dimensional fractional PDEs, Comput. Methods Appl. Mech. Engrg. 400, 2022. 28

    work page 2022

  5. [5]

    W. Guan, K. Yang, Y. Chen, S. Liao, Z. Guan, A dimension-augmented physics informed neural network (DaPINN) with high level accuracy and efficiency, Jour- nal of Computational Physics 491 (2023) 112360

    work page 2023

  6. [6]

    W. He, J. Li, X. Kong, et al., Multi-level physics-informed deep learning for solv- ing partial differential equations in computational structural mechanics, Com- mun. Eng. 3, 151 (2024). https://doi.org/10.1038/s44172-024-00303-3

    doi:10.1038/s44172-024-00303-3 2024

  7. [7]

    Howard, Panos Stinis, SPIKANs: Separable Physics- Informed Kolmogorov-Arnold Networks, arXiv:2411.06286, 2024

    Bruno Jacob, Amanda A. Howard, Panos Stinis, SPIKANs: Separable Physics- Informed Kolmogorov-Arnold Networks, arXiv:2411.06286, 2024. https://doi. org/10.48550/arXiv.2411.06286

    arXiv doi:10.48550/arxiv.2411.06286 2024

  8. [8]

    Namgyu Kang, Jaemin Oh, Youngjoon Hong, Eunbyung Park, PIG: Physics- Informed Gaussians as Adaptive Parametric Mesh Representations, ICLR, 2025

    work page 2025

Show all 33 references
  1. [9]

    T.T. Le, H.T. Dang, T.K. Tran, et al., Necessary and sufficient optimality con- ditions for fuzzy fractional variational problems under granular fuzzy Caputo fractional derivatives, Comput. Appl. Math. 44, 172, 2025. https://doi.org/ 10.1007/s40314-025-03118-w

    2025 doi

  2. [10]

    Mazandarani, N

    M. Mazandarani, N. Pariz, A.V. Kamyad, Granular Differentiability of Fuzzy- Number-Valued Functions, IEEE Trans. Fuzzy Systems, 26(1), 2018

    2018

  3. [11]

    Mazandarani, Y

    M. Mazandarani, Y. Zhao, Z-differential equations, IEEE Trans. Fuzzy Systems, 28, 462–473, 2020

    2020

  4. [12]

    Mazandarani, M

    M. Mazandarani, M. Najariyan, Fuzzy differential equations: conceptual inter- pretations, Evolutionary Intelligence, 17, 441–456, 2024. https://doi.org/10. 1007/s12065-022-00716-z

    2024

  5. [13]

    Muhammad, M

    G. Muhammad, M. Akram, N. Hussain, T. Allahviranloo, Fuzzy Langevin frac- tional delay differential equations under granular derivative, Information Sci- ences, 681, 2024

    2024

  6. [14]

    Paulson, Ali Mesbah, Stochastic physics-informed neural ordinary differential equations, Journal of Computational Physics , 468, 2022

    Jared O’Leary, Joel A. Paulson, Ali Mesbah, Stochastic physics-informed neural ordinary differential equations, Journal of Computational Physics , 468, 2022

    2022

  7. [15]

    Porta, Monica Riva, Alberto Guadagnini, Modeling parametric uncertainty in PDE models via Physics-Informed Neural Networks, Advances in Water Resources, 195, 2025

    Milad Panahi, Giovanni M. Porta, Monica Riva, Alberto Guadagnini, Modeling parametric uncertainty in PDE models via Physics-Informed Neural Networks, Advances in Water Resources, 195, 2025

    2025

  8. [16]

    F. S. Pedro, E. Esmi, L.C. Barros, Calculus for linearly correlated fuzzy function using Fr´ echet derivative and Riemann integral, Information Sciences, 512, Pages 219-237, 2020

    2020

  9. [17]

    https://doi.org/10.48550/ arXiv.2407.18373

    Subhajit Patra, Sonali Panda, Bikram Keshari Parida, et al., Physics- Informed Kolmogorov-Arnold Networks for Dynamical Analysis via Efficient- KAN and WAV-KAN, arXiv:2407.18373v2, 2024. https://doi.org/10.48550/ arXiv.2407.18373

    2024 doi arXiv

  10. [18]

    Piegat, M

    A. Piegat, M. Landowski, Fuzzy Arithmetic Type 1 with Horizontal Mem- bership Functions, in V. Kreinovich (ed.) Uncertainty Modeling , Studies in Computational Intelligence 683, Springer, 2017. https://doi.org/10.1007/ 978-3-319-51052-1_14

    2017

  11. [19]

    Piegat, M

    A. Piegat, M. Landowski, Horizontal Membership Function and Examples of its Applications, International Journal of Fuzzy Systems , 17, 22–30, 2015. https: //doi.org/10.1007/s40815-015-0013-8 29

    2015 doi

  12. [20]

    Raissi, P

    M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations, arXiv preprint arXiv:1711.10561 (2017)

    2017 arXiv

  13. [21]

    Raissi, P

    M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations, arXiv preprint arXiv:1711.10566 (2017)

    2017 arXiv

  14. [22]

    Raissi, P

    M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics 378 (2019) 686–707

    2019

  15. [23]

    Toscano, V

    J.D. Toscano, V. Oommen, A.J. Varghese, et al., From PINNs to PIKANs: recent advances in physics-informed machine learning, Machine Learning in Computational Science & Engineering , 1, 15, 2025. https://doi.org/10.1007/ s44379-025-00015-1

    2025

  16. [24]

    Bohner, E

    Tri Truong, M. Bohner, E. Girejko, et al., Granular fuzzy calculus on time scales and its applications to fuzzy dynamic equations, Information Sciences , 690, 2025

    2025

  17. [25]

    Shupeng Wang, George Em Karniadakis, GMC-PINNs: A general Monte Carlo PINNs method for fractional PDEs on irregular domains, Computer Methods in Applied Mechanics and Engineering , 429, 2024

    2024

  18. [26]

    B. Wang, Y. Li, P. Chen, P. Perdikaris, PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks, Journal of Machine Learning Re- search, 25, 1-51, 2024

    2024

  19. [27]

    S. Wang, H. Wang, P. Perdikaris, On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks, Computer Methods in Applied Mechanics and Engineering 384, 2021

    2021

  20. [28]

    Liu Yang, Xuhui Meng, George Em Karniadakis, B-PINNs: Bayesian Physics- Informed Neural Networks for forward and inverse PDE problems with noisy data, Journal of Computational Physics , 425, 2021

    2021

  21. [29]

    Zadeh, A New Direction in AI: Toward a Computational Theory of Percep- tions, AI Magazine, 22(1), 2001

    L.A. Zadeh, A New Direction in AI: Toward a Computational Theory of Percep- tions, AI Magazine, 22(1), 2001. https://doi.org/10.1609/aimag.v22i1.1545

    2001 doi

  22. [30]

    Zadeh, Toward a generalized theory of uncertainty (GTU) – an outline, Information Sciences, 172, 2005

    L.A. Zadeh, Toward a generalized theory of uncertainty (GTU) – an outline, Information Sciences, 172, 2005

    2005

  23. [31]

    Zadeh, Generalized theory of uncertainty (GTU) – principal concepts and ideas, Computational Statistics & Data Analysis , 51(1), 15-46, 2006

    L.A. Zadeh, Generalized theory of uncertainty (GTU) – principal concepts and ideas, Computational Statistics & Data Analysis , 51(1), 15-46, 2006

    2006

  24. [32]

    Zadeh, A Note on Z-numbers, Information Sciences , 181(8), 2923–2932, 2011

    L.A. Zadeh, A Note on Z-numbers, Information Sciences , 181(8), 2923–2932, 2011

    2011

  25. [33]

    Yuanran Zhu, Yu-Hang Tang, Changho Kim, Learning stochastic dynamics with statistics-informed neural networks,Journal of Computational Physics, 474, 2023. 30

    2023

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.