StableGrad: Backward Scale Control without Batch Normalization
Authors: Jose I. Mestre, Alberto Fernández-Hernández, Cristian Pérez-Corral, Manuel F. Dolz, Enrique S. Quintana-Ortí
Organizations: Universitat Politècnica de València · Universitat Jaume I
Abstract
Training very deep neural networks requires controlling the propagation of magnitudes across depth. Without such control, activations and gradients may vanish, explode, or enter unstable regimes that make optimization fail. Modern architectures often mitigate this problem through Batch Normalization, residual connections, or other normalization layers, which repeatedly re-scale or bypass intermediate representations. However, these mechanisms are not always appropriate. In Physics-Informed Neural Networks (PINNs), the network represents a continuous physical field and its input derivatives define the training objective, making batch-dependent normalization problematic because it can introduce non-local dependencies into the predicted field and its derivatives. We propose StableGrad, an optimizer-level scale-control mechanism that corrects layer-wise weight-gradient imbalances without modifying the forward model. Because the normalization is applied only after backpropagation and before the optimizer update, the network output, its derivatives, and the physical residual remain unchanged. We analyze the effective training dynamics induced by this rescaling and evaluate StableGrad on deep PINNs as the target application, with BatchNorm-free convolutional networks serving as a diagnostic stress test. On PINN benchmarks, StableGrad improves matched-depth solution accuracy and makes deeper models more reliable under standard optimization. On ResNet and EfficientNet architectures, where removing Batch Normalization normally leads to training collapse, StableGrad stabilizes optimization without introducing any other architectural change. These results show that optimizer-level control of weight-gradient scale can provide a practical alternative when forward normalization is unavailable or undesirable.
Normalization renders large parts of neural networks effectively scale invariant, inducing a hidden feedback loop in which learning-rate schedules and weight decay interact through the parameter norm to control the effective step taken by the optimizer. We show that this interaction is governed by an exact discrete-time law: a single scalar quantity captures all schedule and decay forcing, while norm growth induces an opposing geometric self-quenching effect. This yields a sharp boundary that cleanly separates contraction- and expansion-dominated effective learning rate regimes. To understand the underlying mechanism, we provide exact analysis of a fully solved normalized regression model where the dynamics reduce to two dimensions and show that the balance point is intrinsically unstable, implying that constant learning rate with weight decay cannot stably maintain an interior equilibrium and instead produces recurrent behavior driven by discrete-time Jacobian structure. We further extend this perspective across optimizers through unified homogeneous-optimizer framework that reveals a structural dichotomy in self-quenching strength, providing a first-principles explanation for why adaptive methods exhibit systematically weaker stabilization under normalization. Across dynamical systems and neural networks (MLP, CNN, GPT2 / MNIST, CIFAR, wikiText, OpenWebText), the predicted law holds with high precision and enables direct control of training via the identified scalar, with performance peaking sharply at the predicted boundary. Together, these results isolate a single governing quantity for scale-invariant optimization, providing a precise and actionable lens on training dynamics, optimizer behavior, and schedule design in modern deep learning. Code is available in https://github.com/shasanamin/normalized-optimization-dynamics.
While Physics-Informed Neural Networks (PINNs) are powerful for solving Partial Differential Equations (PDEs), their training is often paralyzed by gradient pathology. The gradients from the PDE residuals and boundary constraints oppose each other, trapping the model in local minima. Current solutions, such as adaptive weighting or hard constraints, either fail to fundamentally resolve this ill-conditioning or are limited to simple geometries. In this study, we systematically analyze the possible causes of this gradient pathology from the perspectives of loss landscapes and optimization dynamics. Based on the obtained conclusion, we propose Constraint-Aligned loss with Manifold Lifting (CAML). By reformulating all zeroth-order terms into aligned constraints, our method effectively mitigates gradient conflicts. In addition, we introduce a delay factor to help the optimizer skip the high-curvature area. Experiments demonstrate that our CAML significantly enhances numerical stability and efficiency in highly complex PINN problems. Our code is open-sourced on https://github.com/YichenLuo-0/CAML.
Physics-Informed Neural Networks (PINNs) often suffer from slow convergence, training instability, and reduced accuracy on challenging partial differential equations due to the anisotropic and rapidly varying geometry of their loss landscapes. We propose a lightweight curvature-aware optimization framework that augments existing first-order optimizers with an adaptive predictive correction based on secant information. Consecutive gradient differences are used as a cheap proxy for local geometric change, together with a step-normalized secant curvature indicator to control the correction strength. The framework is plug-and-play, computationally efficient, and broadly compatible with existing optimizers, without explicitly forming second-order matrices. Experiments on diverse PDE benchmarks show consistent improvements in convergence speed, training stability, and solution accuracy over standard optimizers and strong baselines, including on the high-dimensional heat equation, Gray--Scott system, Belousov--Zhabotinsky system, and 2D Kuramoto--Sivashinsky system.