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arxiv: 2605.05217 · v1 · submitted 2026-04-17 · 💻 cs.LG · cs.AI

Physics-Informed Neural Networks with Learnable Loss Balancing and Transfer Learning

Reza Pirayeshshirazinezhad This is my paper

Pith reviewed 2026-05-10 09:23 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords physics-informed neural networkslearnable loss balancingtransfer learningadaptive weightingdata scarcityheat transfer predictionscientific machine learning
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The pith

A learnable blending neuron in PINNs dynamically balances physics and data terms based on uncertainties for improved performance with scarce data.

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

This paper introduces an adaptive physics-informed neural network that uses a learnable blending neuron to automatically adjust the importance of physics residuals versus data losses according to their uncertainties. The goal is to eliminate manual tuning of loss weights, which often destabilizes training in traditional PINNs, especially when data is limited. Transfer learning is added to leverage pre-trained representations from similar problems and fine-tune on new ones with minimal data. In a test case predicting heat transfer in miniature heat sinks using just 87 data points from CFD simulations, the method achieves less than 8% error while surpassing shallow networks, kernel methods, and physics-only models. Readers might care because this offers a more robust way to incorporate physical laws into machine learning for scientific problems where gathering data is costly.

Core claim

The authors establish that replacing fixed or heuristic loss weights in PINNs with a learnable blending neuron enables the network to dynamically set the relative contributions of physics-based and data-driven terms according to their uncertainties. Paired with a transfer learning strategy that reuses domain representations, this framework delivers stable training and accurate predictions on data-scarce tasks, such as heat transfer modeling in liquid-metal heat sinks where errors stay below 8%.

What carries the argument

The learnable blending neuron, a component that computes dynamic weights for the combined loss function by considering the uncertainties of individual physics and data loss terms.

Load-bearing premise

That the learnable blending neuron will produce stable training and superior generalization on new physical systems rather than overfitting to the uncertainty patterns of the 87-point heat transfer dataset.

What would settle it

Demonstrating that when applied to a distinct physical system with different data characteristics, the adaptive PINN exhibits unstable training or errors exceeding those of fixed-weight baselines.

Figures

Figures reproduced from arXiv: 2605.05217 by Reza Pirayeshshirazinezhad.

Figure 1
Figure 1. Figure 1: Schematic of the proposed self-supervised adaptive PINN with transfer learning. The [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Effect of transfer learning from different layers. The first-layer transfer achieves the lowest [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: KDE comparison of water and sodium Nusselt numbers. Sodium exhibits higher variance [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Distribution of the learned physics coefficient neuron in the self-supervised PINN, cen [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: GP and SVR predictions on holdout dataset. SVR–Bayesian achieves better fit than GP, [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Neural network predictions on holdout dataset. The self-supervised PINN provides the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original abstract

We propose a self-supervised physics-informed neural network (PINN) framework that adaptively balances physics-based and data-driven supervision for scientific machine learning under data scarcity. Unlike prior PINNs that rely on fixed or heuristic weighting of physics residuals and data loss, our approach introduces a learnable blending neuron that dynamically adjusts the relative contribution of each term based on their uncertainties. This mechanism enables stable training and improved generalization without manual tuning. To further enhance efficiency, we integrate a transfer learning strategy that reuses representations from related domains and adapts them to new physical systems with limited data. We validate the framework for the prediction of heat transfer in liquid-metal miniature heat sinks using only 87 CFD datapoints, where the adaptive PINN achieves an error <8%, outperforming shallow neural networks, kernel methods, and physics-only baselines. Our framework provides a general recipe for embedding physics adaptively into neural networks, offering a robust and reproducible approach for data-scarce problems across various scientific domains, including fluid dynamics and material modeling.

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

3 major / 0 minor

Summary. The manuscript proposes a self-supervised PINN framework that uses a learnable blending neuron to dynamically balance physics residuals and data losses according to estimated uncertainties, augmented by transfer learning to adapt representations across physical systems under data scarcity. It validates the approach on heat-transfer prediction for liquid-metal miniature heat sinks using only 87 CFD points, reporting error below 8% and outperformance over shallow neural networks, kernel methods, and physics-only baselines.

Significance. If the central claims hold after addressing the noted gaps, the work would contribute a practical mechanism for automating loss weighting in PINNs, a persistent practical challenge, while the transfer-learning component offers a route to leverage related domains for new systems. The emphasis on a small, engineering-relevant dataset is a strength for demonstrating applicability in data-scarce scientific ML; credit is due for attempting to ground the blending in uncertainty rather than heuristics.

major comments (3)
  1. [Abstract] Abstract: the performance claim of error <8% and outperformance is presented without details on network architecture, the internal structure or uncertainty estimation inside the blending neuron, baseline implementations, cross-validation procedure, or error bars, rendering the central empirical claim impossible to assess.
  2. [Methods] Methods: no mathematical formulation, pseudocode, or architectural diagram specifies how the blending neuron computes or uses uncertainties to set weights (as opposed to learning arbitrary coefficients); the added free parameters therefore risk reducing to a fitted weighting scheme on the 87-point dataset rather than enforcing the claimed uncertainty-based mechanism.
  3. [Experiments] Experiments/Results: validation is restricted to a single heat-transfer task plus transfer learning; no ablation isolates the blending neuron's contribution from the transfer-learning component or from a standard PINN, and no additional physical systems are tested to support the generalization claim without manual tuning.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment point by point below, indicating where we agree and the revisions we will implement.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the performance claim of error <8% and outperformance is presented without details on network architecture, the internal structure or uncertainty estimation inside the blending neuron, baseline implementations, cross-validation procedure, or error bars, rendering the central empirical claim impossible to assess.

    Authors: We agree that the abstract is too concise to allow full assessment of the central claims. In the revised manuscript, we will expand the abstract with a brief description of the network architecture, the blending neuron's internal structure for uncertainty estimation, the baseline implementations, the cross-validation procedure, and the inclusion of error bars on reported errors. This will improve assessability without substantially increasing length. revision: yes

  2. Referee: [Methods] Methods: no mathematical formulation, pseudocode, or architectural diagram specifies how the blending neuron computes or uses uncertainties to set weights (as opposed to learning arbitrary coefficients); the added free parameters therefore risk reducing to a fitted weighting scheme on the 87-point dataset rather than enforcing the claimed uncertainty-based mechanism.

    Authors: This is a fair critique of the current presentation. The manuscript describes the blending neuron at a conceptual level but lacks the requested explicit details. In the revision, we will add the mathematical formulation showing how the neuron uses estimated uncertainties to compute weights, pseudocode for the loss balancing procedure, and an architectural diagram. These additions will demonstrate that the mechanism is uncertainty-driven rather than an arbitrary fit. revision: yes

  3. Referee: [Experiments] Experiments/Results: validation is restricted to a single heat-transfer task plus transfer learning; no ablation isolates the blending neuron's contribution from the transfer-learning component or from a standard PINN, and no additional physical systems are tested to support the generalization claim without manual tuning.

    Authors: We acknowledge that the experimental section would be strengthened by ablations and broader testing. We will add ablation experiments in the revision to isolate the blending neuron's contribution from both transfer learning and a standard PINN baseline. However, extending validation to multiple additional physical systems would require new datasets and experiments outside the current scope; we will instead expand the discussion to explain how the transfer learning results already demonstrate adaptation without manual tuning on the target task. revision: partial

standing simulated objections not resolved
  • Requirement to test on multiple additional physical systems beyond the current heat-transfer task and transfer learning setup, as this would necessitate substantial new data collection and experiments not performed in the original work.

Circularity Check

0 steps flagged

No circularity in derivation chain; results grounded in external data

full rationale

The paper defines an architectural extension (learnable blending neuron for loss weighting plus transfer learning) and reports empirical performance on an independent 87-point CFD heat-transfer dataset. No equations or claims are shown that algebraically reduce the reported error or generalization improvement to the model inputs by construction. The validation uses external benchmarks (shallow NNs, kernel methods, physics-only baselines) rather than internal self-consistency loops. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard PINN assumptions plus two new elements whose behavior is not independently verified outside the reported experiment.

free parameters (1)
  • blending neuron parameters
    Weights inside the learnable neuron are optimized during training to set the relative loss contributions.
axioms (2)
  • standard math Physics residuals can be evaluated via automatic differentiation inside the neural network
    Core assumption of all PINNs; invoked when the physics loss term is formed.
  • domain assumption Representations learned on related physical domains transfer usefully to the target heat-transfer problem
    Required for the transfer-learning component to reduce data needs.
invented entities (1)
  • learnable blending neuron no independent evidence
    purpose: Dynamically compute loss weights from estimated uncertainties of physics and data terms
    New architectural component introduced to replace fixed or heuristic weighting.

pith-pipeline@v0.9.0 · 5470 in / 1530 out tokens · 48560 ms · 2026-05-10T09:23:24.953268+00:00 · methodology

discussion (0)

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Reference graph

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