{"id":"aed067ed-73a7-463c-b739-5018338aa6bc","arxiv_id":"2505.03806","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The paper defines Perception-Informed Neural Networks as a unifying framework in which physics laws and fuzzy or probabilistic expert perceptions enter neural networks through loss functions, and proposes new variants such as MOEINNs, TKINNs, and FINNs without empirical validation.","lead":"This paper introduces Perception-Informed Neural Networks, a conceptual framework that adds perception-based information such as fuzzy rules, intervals, and probability statements to neural network loss functions, extending Physics-Informed Neural Networks. It is a framework and taxonomy paper, with illustrations but no experiments, so its practical value is not yet demonstrated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proposed sureness loss (Eqs. 27, 33) is mathematically ill-posed: N_gi is a probability density, not a probability, so sureness can exceed 1 and the loss can push residuals away from the expert constraint.","rationale":"The paper is best read as a conceptual taxonomy plus a set of proposed loss constructions. Its central claim, that any perception-based constraint can be integrated by precisiation followed by a differentiable penalty, stands or falls on whether the proposed penalties are faithful to the constraints. The sureness loss is not faithful: at the paper's own sigma = 0.01, the normal density peaks at about 39.89, so 'sureness' can exceed 1 by a factor of about 40, and minimizing (1 - sureness)^2 becomes a regression to a target density value rather than to the expert's constraint. This is not merely a missing convergence analysis; the objective has the wrong minima even in the zero-residual limit. The reader identified the general weakness that the loss forms are asserted without stability or convergence support; my reading agrees and sharpens it to a concrete mathematical error in Eqs. (27) and (33). The taxonomy itself, including the recasting of PINNs as the singular precisiation mode, is reasonable and could survive a corrected sureness definition, so I do not move the verdict. I would keep the conditional acceptance but require the authors to replace densities by probabilities, re-examine the boundedness of the fuzzy-graph loss in Eq. (34), and demonstrate the corrected losses on the paper's own examples.","tokens_in":20279,"tokens_out":7819,"duration_ms":87955,"concrete_test":"On the damped-oscillator example of Eqs. (28)-(33), set sigma = 0.01 and take the exact modal solution so the residual g = 0 at a collocation point. Compute L_s = (1 - 39.89*mu)^2 for mu in [0, 1]: the minimum occurs near mu = 0.025 with L_s ≈ 0, while mu = 1 gives L_s ≈ 1512. Then take one gradient step of L_s with respect to the network outputs and check whether the PDE residual |g| increases. If it does, Eq. (33) is not enforcing the expert's constraint and the sureness loss must be redefined using a probability mass or a properly normalized measure, not a density.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central mechanism of PrINNs is to convert a precisiated perception constraint into a differentiable loss penalty. For the SINNet/TKINN contribution, that penalty is L_s = (1 - sureness)^2 in Eq. (27), where Eq. (33) defines sureness = mu * N_gi. Here N_gi is the value of the normal density in Eq. (32), not a probability. A normal density is not bounded above by 1: its maximum is 1/(sigma * sqrt(2*pi)), which for the paper's own example sigma = 0.01 is about 39.89. Therefore sureness lies in [0, 39.89], not [0, 1], and L_s is minimized at sureness = 1 rather than at the expert's modal constraint. At the exact solution with residual g = 0, N_gi ≈ 39.89, so L_s is minimized by mu ≈ 0.025 instead of mu = 1. Equivalently, with mu fixed near 1, the loss drives the residual to the contour where the density equals 1/mu, displacing the solution away from the value the experts specified. The loss thus does not implement the stated 'impossibility implies improbability' principle. Because the paper provides no experiments or convergence analysis for any of the new losses, this concrete failure in a flagship proposed loss leaves the central 'add the constraint to the loss' mechanism unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20591,"tokens_out":9301,"duration_ms":89200,"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":[{"comment":"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.","section":"§3.1, Eqs. (27), (32), (33)"},{"comment":"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.","section":"§3.2, Eq. (34) and Eqs. (35), (37), (39)"},{"comment":"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.","section":"Abstract and §3.2"},{"comment":"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.","section":"§3.1, Eq. (24)"}],"minor_comments":[{"comment":"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.","section":"§2.1, Eq. (6)"},{"comment":"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).","section":"§2.2, Eq. (26)"},{"comment":"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.","section":"§2.1, Eq. (7)"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is best viewed as a conceptual framework paper. The mathematical flaw in the sureness loss (Eqs. 27, 32, 33) is load-bearing for the TKINN/SINNet contribution and should be corrected. In addition, the journal should consider whether the lack of any experiments or convergence analysis is acceptable for a framework paper; the current abstract makes rather strong claims that the body does not substantiate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis is a conceptual framework paper, not an empirical study. The new thing here is a taxonomy: PINNs and many of their variants are recast as different \"modes of precisiation\" of perception, following Zadeh's GTU. That framing is genuinely useful — it puts crisp, fuzzy, probabilistic, interval, and mixed-expert informed networks on a common footing. The named loss forms (FINNs' M(1-R)^2, FcINNs' M^(1-µ) weighting, SINNets' sureness loss) are new as combinations, and the ANFIS comparison is informative.\n\nWhat the paper doesn't do is demonstrate any of it. There are no experiments, no convergence analysis, no baselines. Claims about online control, solving IDEs, discovering new differential equations are asserted, not shown. As a position paper that's acceptable if framed as a roadmap; here they're presented as contributions.\n\nThe more concrete problem is the sureness loss. Sureness is defined as µ × N_gi, where N_gi is a normal density evaluated at the residual. A density is not bounded by 1; in their own example (variance 0.01), the peak is about 3.99. So sureness can exceed 1, and minimizing (1 - sureness)^2 will not drive the residual to the expert's modal value. At g=0 the loss is minimized by µ ≈ 0.25 rather than µ=1. Equivalently, with µ fixed near 1, the loss pushes the residual to a contour away from the mode. The loss simply does not implement \"impossibility implies improbability.\" This is a load-bearing flaw in one of the three headline contributions.\n\nThe rest of the algebra is mostly fine, and the fuzzy-graph and possibility-weighted losses are plausible. But the central assumption — that a precisiated perception reliably becomes a trainable loss — is unvalidated. The paper also leans heavily on the authors' own prior constructs (IDEs, ZDEs, granular derivatives), which limits immediate uptake.\n\nMy take: the taxonomy is worth discussing, and a serious referee should engage with it. But the overclaiming and the sureness error mean the paper needs major revision before it can be trusted as a technical contribution. I'd want either experiments or a clear reduction in scope, plus a corrected or removed sureness loss.\n\nRecommendation: send to peer review, expect heavy revision.","headline":"A useful taxonomy of informed neural networks, but the sureness loss is mathematically flawed and the paper overclaims with zero experiments.","tokens_in":21174,"tokens_out":6486,"would_cite":false,"duration_ms":59170,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Artificial Intelligence","Machine Learning","Computational Science and Engineering","Generalized Theory of Uncertainty","Deep Neural Networks","Perception Computing","Dynamical Systems","Physics-Informed Neural Networks"],"falsifier":"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.","tokens_in":20004,"feed_emoji":"🧠","tokens_out":9599,"duration_ms":89534,"temperature":0.7,"pith_summary":"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.","feed_headline":"One training recipe unifies physics and fuzzy-rule neural nets","feed_subtitle":"The paper frames crisp, probabilistic, possibilistic, and fuzzy-rule networks as one precisiation mode added to the loss.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the PINN loss construction that PrINNs generalize; the singular-mode case is read as a special case.","marker":"[20, 21, 22]"},{"why":"Supplies the generalized theory of uncertainty and the modes of precisiation that define the PrINN recipe.","marker":"[30, 31]"},{"why":"Defines imprecise differential equations and the conceptual role of perception-based propositions that PrINNs are built to handle.","marker":"[12]"},{"why":"Gives horizontal membership functions, the machinery used to turn fuzzy residuals into granular loss terms.","marker":"[18, 19]"},{"why":"Introduces Z-differential equations, the mixture-of-modes construction that motivates sureness-informed networks.","marker":"[11]"},{"why":"Defines Z-numbers and the product-based notion of sureness used in the transformed-knowledge loss.","marker":"[32]"},{"why":"Previous interval and fuzzy physics-informed neural networks serve as the possibility-mode case that PrINNs absorb.","marker":"[3]"},{"why":"PINN under uncertainty supplies the probability-mode example that the paper re-describes within PrINNs.","marker":"[15]"}],"fun_headline_variants":["PrINNs: One loss recipe for physics and fuzzy rules","Perception-informed nets extend physics-informed ones","From crisp to fuzzy: Unifying informed neural nets","Solve fuzzy differential equations with PrINNs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["PrINNs: One loss recipe for physics and fuzzy rules","Perception-informed nets extend physics-informed ones","From crisp to fuzzy: Unifying informed neural nets","Solve fuzzy differential equations with PrINNs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000174,"raw_usage":{"total_tokens":1312,"prompt_tokens":1002,"completion_tokens":310,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":249}},"tokens_in":618,"tokens_out":310,"duration_ms":3765,"temperature":1.0,"reasoning_tokens":249,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:26:23.472248+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Mazandarani, M","cited_arxiv_id":null,"evidence_quote":"Defines imprecise differential equations and the conceptual role of perception-based propositions that PrINNs are built to handle."},{"cited_title":"Mazandarani, Y","cited_arxiv_id":null,"evidence_quote":"Introduces Z-differential equations, the mixture-of-modes construction that motivates sureness-informed networks."},{"cited_title":"Zadeh, A Note on Z-numbers, Information Sciences , 181(8), 2923–2932, 2011","cited_arxiv_id":null,"evidence_quote":"Defines Z-numbers and the product-based notion of sureness used in the transformed-knowledge loss."},{"cited_title":"Fuhg, Ioannis Kalogeris, Am´ elie Fau, Nikolaos Bouklas, Interval and fuzzy physics-informed neural networks for uncertain fields, Probabilistic Engineering Mechanics, 68, 2022","cited_arxiv_id":null,"evidence_quote":"Previous interval and fuzzy physics-informed neural networks serve as the possibility-mode case that PrINNs absorb."},{"cited_title":"Porta, Monica Riva, Alberto Guadagnini, Modeling parametric uncertainty in PDE models via Physics-Informed Neural Networks, Advances in Water Resources, 195, 2025","cited_arxiv_id":null,"evidence_quote":"PINN under uncertainty supplies the probability-mode example that the paper re-describes within PrINNs."}],"review_version":1}