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Harmonized Cone for Feasible and Non-conflict Directions in Training Physics-Informed Neural Networks

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Physics-Informed Neural NetworksMulti-loss OptimizationGradient Conflict ResolutionFeasible DirectionsNonconvex Convergence
Abstract:

Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving PDEs, yet training is difficult due to a multi-objective loss that couples PDE residuals, initial/boundary conditions, and auxiliary physics terms. Existing remedies often yield infeasible scaling factors or conflicting update directions, resulting in degraded performance. In this paper, we show that training PINNs requires jointly considering feasible scaling factors and a non-conflict direction. Through a geometric analysis of per-loss gradients, we define the harmonized cone\textit{harmonized cone} as the intersection of their primal and dual cones, which characterizes directions that are simultaneously feasible and non-conflicting. Building on this, we propose HARMONICHARMONIC (HARMONIzed Cone gradient descent), a training procedure that computes updates within the harmonized cone by leveraging the Double Description method to aggregate extreme rays. Theoretically, we establish convergence guarantees in nonconvex settings and prove the existence of a nontrivial harmonized cone. Across standard PDE benchmarks, HARMONICHARMONIC generally outperforms state-of-the-art methods while ensuring feasible and non-conflict updates.

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Overview

Overall Novelty Assessment

The paper introduces a harmonized cone framework for balancing multiple loss terms in PINN training, combining feasible scaling factors with non-conflicting gradient directions. It resides in the Adaptive Weight Adjustment Mechanisms leaf, which contains nine papers addressing automatic tuning of loss weights during training. This leaf sits within the broader Loss Balancing and Weighting Strategies branch, indicating a moderately crowded research direction focused on dynamic weight adjustment rather than static schemes or multi-objective formulations.

The taxonomy reveals neighboring leaves for Multi-Objective Optimization Frameworks (six papers treating PINN training as Pareto trade-offs) and Dimensional Analysis approaches (two papers deriving weights from physical units). The paper's geometric cone-based method diverges from gradient magnitude heuristics common in sibling works and from Pareto-based methods in the adjacent leaf. The exclude_note clarifies that gradient pathology mitigation belongs under Training Strategies, suggesting the harmonized cone's dual focus on feasibility and conflict resolution may bridge multiple categories.

Among eleven candidates examined, the harmonized cone concept itself encountered no refutable prior work, while the HARMONIC algorithm examined one candidate with no clear overlap. The theoretical convergence guarantees examined ten candidates and found four potentially refutable matches, indicating this contribution has more substantial prior work in nonconvex optimization theory. The limited search scope (eleven total candidates from semantic search) means these statistics reflect top-ranked matches rather than exhaustive coverage, so contributions appearing novel here may still have unexamined precedents.

Based on the top-eleven semantic matches, the geometric harmonization approach appears relatively fresh within adaptive weighting mechanisms, though the theoretical analysis overlaps with existing convergence literature. The taxonomy structure suggests the field is actively exploring diverse balancing strategies, and this work's cone-based geometry offers a distinct angle compared to gradient-norm or uncertainty-driven siblings.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
11
Contribution Candidate Papers Compared
4
Refutable Paper

Research Landscape Overview

Core task: training physics-informed neural networks with multiple loss terms. Physics-informed neural networks (PINNs) embed governing equations and boundary conditions as soft constraints in the loss function, but balancing these multiple terms during training remains a central challenge. The field has organized itself into several main branches. Loss Balancing and Weighting Strategies focus on adaptive mechanisms that dynamically adjust term weights to prevent one component from dominating the optimization, as seen in works like Self-Adaptive Balanced[24] and Adaptive Loss Weighting[10]. Loss Function Design and Formulation explores alternative formulations and regularization schemes, such as Modified Loss Function[4] and Gradient Enhanced PINN[1], to improve convergence and accuracy. Training Strategies and Optimization Techniques address broader algorithmic improvements, including multi-scale frameworks and novel optimizers. Domain-Specific PINN Applications demonstrate how these methods perform in concrete settings like fluid dynamics, solid mechanics, and electromagnetics, while Foundational Reviews and Methodological Surveys, including Comprehensive Review[7] and Loss Function Design Review[8], synthesize emerging best practices across the landscape. A particularly active line of work centers on adaptive weight adjustment mechanisms, where methods automatically tune loss term coefficients based on gradient magnitudes, residual imbalances, or uncertainty estimates. Harmonized Cone[0] sits squarely within this branch, proposing a cone-based geometric approach to harmonize competing gradients from different loss components. This contrasts with gradient-based heuristics in Loss Attentional[6] and uncertainty-driven schemes in Weighted Uncertainty[36], which rely on different signals to guide weight updates. Another vibrant area involves multi-objective optimization perspectives, as in Multi-Objective Loss Balancing[5], which frames the problem as a Pareto trade-off rather than a single scalar objective. Open questions persist around the interplay between loss weighting and network architecture choices, the sensitivity of adaptive schemes to hyperparameters, and the extent to which domain-specific physics should inform the balancing strategy. Harmonized Cone[0] addresses these concerns by offering a geometrically motivated criterion that aims to reduce manual tuning while maintaining stable training across diverse PDE types.

Claimed Contributions

Harmonized cone concept for PINN training

The authors introduce the harmonized cone, defined as the intersection of primal and dual cones of per-loss gradients. This geometric construct characterizes update directions that are both feasible (representable as nonnegative combinations of loss gradients) and non-conflicting (ensuring no loss increases).

0 retrieved papers
HARMONIC training algorithm

The authors propose HARMONIC, a gradient-based training procedure that ensures updates remain within the harmonized cone. The method uses the Double Description method to convert half-space representations into vertex representations and aggregates extreme rays to form feasible and non-conflict update directions.

1 retrieved paper
Theoretical convergence guarantees and existence proof

The authors provide theoretical analysis showing that HARMONIC converges to Pareto-stationary points in nonconvex settings at a rate of O(1/√T). They also prove that a nontrivial harmonized cone always exists, ensuring the method is applicable across all training scenarios.

10 retrieved papers
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Harmonized cone concept for PINN training

The authors introduce the harmonized cone, defined as the intersection of primal and dual cones of per-loss gradients. This geometric construct characterizes update directions that are both feasible (representable as nonnegative combinations of loss gradients) and non-conflicting (ensuring no loss increases).

Contribution

HARMONIC training algorithm

The authors propose HARMONIC, a gradient-based training procedure that ensures updates remain within the harmonized cone. The method uses the Double Description method to convert half-space representations into vertex representations and aggregates extreme rays to form feasible and non-conflict update directions.

Contribution

Theoretical convergence guarantees and existence proof

The authors provide theoretical analysis showing that HARMONIC converges to Pareto-stationary points in nonconvex settings at a rate of O(1/√T). They also prove that a nontrivial harmonized cone always exists, ensuring the method is applicable across all training scenarios.

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