Learning the Constitutive Behavior of Materials via Neural Operators and Causal Attention: Case Studies in Plasticity and Damage

Classical constitutive modeling of path-dependent inelastic materials relies on internal state variables whose evolution equations must be postulated based on domain knowledge and calibrated against experimental data. However, in many practical settings, the relevant internal variables are typically not measurable in experiments, and the constitutive response must be inferred entirely from measured strain-stress data without any prior knowledge of the material's internal state. We propose a data-driven constitutive modeling framework based on the concept of a material operator, which treats a deforming material as a functional mapping from its entire strain history to the corresponding stress response. In contrast to traditional autoregressive or recurrent formulations, the model is trained directly on full loading paths as function-to-function mappings, predicting complete stress trajectories in a single parallel forward pass. Temporal path dependence is enforced through a causally masked attention mechanism embedded within the operator, which restricts the model's attention to past material states while preserving computational parallelizability. Spectral convolutions provide discretization-invariant representations in the frequency domain, while causal attention captures highly adaptive, non-local history dependence. Furthermore, sinusoidal activation functions are used to resolve the strong nonlinear transitions inherent in inelastic regimes. The framework is evaluated across multidimensional, rate-independent material models exhibiting complex phenomena, with an emphasis on nonlinear plasticity and ductile damage accumulation. The results demonstrate accurate and robust predictions of irreversible deformation mechanisms while simultaneously achieving resolution invariance and excellent parallel efficiency.

Dimensionless Controls of Plasticity Under Alternating Tasks: From Evolutionary Biology to Continual Learning

Plasticity under changing environments is central to both evolutionary biology and continual learning. Motivated by recent work on genotype--phenotype maps, we study a minimal deep-learning analogue where a network is trained alternately on two Boolean label sets, and ask which biological controls of plasticity survive the translation to gradient descent. Reinterpreting four proposed biological factors as quantities of training dynamics, we find the system reduces to two dimensionless controls: the task disagreement $r$, the fraction of disagreeing labels, and the reach $ηT$, the product of learning rate and switching period. We derive two bounds on plasticity: $r$ alone fixes an extremal geometric floor on the utopia distance, while $r$ and $ηT$ jointly bound forgetting. Across 9,720 trajectories, an ANOVA confirms that $r$, $η$, and $T$ dominate, while the effect of neutral-set size (emphasized in the biological setting) is negligible. The optimal reach itself follows an approximate inverse power law $ηT^{*}\propto r^{-1.18}$, yielding a heuristic that sets the optimal reach $ηT^*$ from the task disagreement alone. The analogy that survives is therefore dynamical rather than geometric, and our setting enables a view of plasticity through the lens of other driven systems in physics and engineering.

Forgetting, plasticity, and co-observation: a third facet of continual learning

Efficient continual learning remains a fundamental challenge for deep neural networks. While catastrophic forgetting and loss of plasticity are widely considered the primary obstacles to overcome, we show that these two issues cannot fully explain the performance gap between naive sequential training and offline joint training. In this paper, we highlight data co-observation as a distinct factor influencing continual learning performance. By decoupling the constraints of separate data access from stability and plasticity, we systematically investigate the representational benefits gained by observing training data together. Empirically, we demonstrate a consistent performance difference between joint and separate training across both supervised and self-supervised paradigms in generic data-incremental "chunking" scenarios, whilst mitigating forgetting and controlling for plasticity. Our findings indicate that simultaneous observation of training data (co-observation) yields benefits to the learner's generalization that extend well beyond mere knowledge retention, and that this effect does not require a specific continual distribution shift. Furthermore, we contextualize prominent continual learning mechanisms through this lens: while distillation-based approaches act only as effective knowledge retention mechanisms, our results suggest that the empirical success of memory replay goes beyond the mitigation of forgetting, actively reintroducing the benefits of data co-observation into the learning process.

SingularClip: Preventing Spectral Collapse to Maintain Plasticity in Continual and Reinforcement Learning

Neural networks trained on nonstationary tasks frequently lose the ability to fit new targets, a phenomenon referred to as loss of plasticity. We identify a novel source of plasticity loss due to the growing anisotropy of weight matrices' singular values during training, and analyze this phenomenon both empirically and theoretically. To mitigate this issue, we introduce SingularClip, a procedure that periodically clips the singular values of all weight matrices. We show that SingularClip performs strongly against baselines across a range of tasks in both continual supervised learning and deep reinforcement learning.

NeuMoSync: End-to-End Neuromodulatory Control for Plasticity and Adaptability in Continual Learning

Continual learning (CL) requires models to learn tasks sequentially, yet deep neural networks often suffer from plasticity loss and poor knowledge transfer, which can impede their long-term adaptability. Drawing high-level inspiration from global neuromodulatory mechanisms in the brain, we introduce Neuromodulation and Synchronization (NeuMoSync), a novel architecture that integrates dynamic, neuron-specific modulation into deep neural networks to enhance their adaptability and plasticity. NeuMoSync extends standard neural network architectures with learnable feature vectors for each neuron that track network-wide historical context and with a module operating at a higher level of abstraction. This module synthesizes neuron-specific signals, conditioned on both current inputs and the network's evolving state, to adaptively regulate activation dynamics and synaptic plasticity. Evaluated on diverse CL benchmarks, including memorization (Random Label CIFAR-10 and Random Label MNIST), concept drift (Shuffle CIFAR-10 and Shuffle Mini-ImageNet), class-incremental learning (Class Split ImageNet and Class Split CIFAR-100), and domain-incremental learning (Permuted MNIST), NeuMoSync demonstrates strong performance in retaining plasticity and achieves improvements in both forward and backward adaptation compared with existing methods. Ablation studies validate the necessity of each component, while analysis of the learned modulatory signals reveals interpretable coordination patterns across tasks. Our work underscores the potential of integrating global coordination mechanisms into deep learning systems to advance robust, adaptive continual learning. The code is publicly available at https://github.com/RoozbehRazavi/NeuMoSync.

Plasticity of Growing and Elastic Neural Networks in Online Continual Learning

Neural networks that can grow or both grow and shrink during learning, referred to as growing neural networks and elastic neural networks, respectively, have recently been explored in offline continual learning with a particular focus on catastrophic forgetting. Driven by the observations that 1) online continual learning closely resembles how animals learn; 2) loss of plasticity---the progressive decline in a learning network's ability to learn---is another crucial challenge facing continual learning; and 3) incremental introduction of randomly initialized hidden units was recently shown to help preserve plasticity, in this paper, we study the plasticity of several foundational growing and elastic networks in online continual learning. Our experiments in supervised learning settings show that adaptive growing networks, which incrementally incorporate new, randomly initialized units to the network while keeping all existing connections adaptive, can maintain high prediction accuracy without losing plasticity despite the continuous increase in the dead hidden unit proportion. Furthermore, we demonstrate that adaptive elastic networks, which in addition to progressively adding new hidden units also prune estimated dead hidden units at the beginning of each new task, can achieve excellent accuracy without loss of plasticity while simultaneously maintaining a near-constant, compact size. Our results suggest that growing and elastic networks, which exhibit the ability to adapt its structure to the relevant learning objectives, can be a promising class of algorithms also for preserving high plasticity in online continual learning.

Calibrated Partial Resets: Preventing Policy Collapse in Continual Reinforcement Learning

Neural networks are hindered by accumulating dormant neurons and loss of expressivity throughout training, particularly in non-stationary data settings, such as continual supervised and reinforcement learning. Recently, neuron resets have been used to maintain gradient flow and restore plasticity. However, full unit reinitialization often sacrifices peak performance and can destabilize training, leading to policy collapse. To preserve plasticity without destabilizing training, we propose Calibrated Partial Resets (CPR), an optimizer that periodically pulls low-utility neurons toward their initialization, with pull strength scaled by each neuron's utility. Unlike binary reset methods, partial resets avoid brittleness; unlike uniform decay, calibrated utility-scaling concentrates adjustment on the units that need it most. Among compared methods, only CPR avoids policy collapse over 400M training steps in SlipperyAnt, and it outperforms prior decay and reset-based methods on Continual MetaWorld and Continual MinAtar benchmarks. Ablations reveal a tunable trade-off between plasticity and peak performance, highlighting utility-scaled reinitialization as a promising direction for continual learning.

Learning to Access Computation: Accessibility Plasticity as a Principle of Adaptive Intelligence

Modern neural networks primarily adapt through parameter modification within predefined computational structures. While recent methods introduce modularity, conditional computation, and parameter-efficient adaptation, they generally do not distinguish computational capability from computational accessibility as separate adaptive variables. This work introduces Accessibility Plasticity, a principle of adaptive computation in which systems adapt not only by changing what computation exists, but also by reorganizing which existing computations can interact and participate. We formalize Accessibility Plasticity through a relationship-based operational realization and establish a reuse-first hierarchy of adaptation, where accessibility modification precedes more costly capability and structural changes. A proof-of-concept evaluation on sequential learning tasks shows that accessibility adaptation can reduce capability modification while maintaining comparable task performance. These results suggest accessibility as a distinct adaptive dimension and provide a foundation for future dynamic neural systems whose computational relationships evolve with changing environments.

Local Redundancy: An Information-Theoretic Measure of Plasticity from Synthetic Memorization

Plasticity -- a neural network's ability to adapt to new tasks -- is critical for continual and transfer learning. Existing measures, such as effective rank, dead neuron fraction, and weight norm, lack theoretical grounding and correlate poorly with performance on new tasks. We introduce local redundancy, an information-theoretic measure derived from universal compression theory. We define local redundancy as the worst-case redundancy of a local model family -- parameters in an infinitesimal neighborhood along gradient directions -- and show this is a principled measure of plasticity. Although local redundancy is intractable to compute exactly, we prove that the expected squared gradient norm on a synthetic memorization task provides an efficiently computable lower bound. Experiments on continual image classification and time series transfer learning demonstrate that local redundancy predicts downstream performance better than existing measures and enables pretraining checkpoint selection where validation loss plateaus.

Physics-Informed Discovery of Yield Functions in Plasticity via Convex Neural Representations

Identifying anisotropic yield functions remains challenging since yielding is not directly observed in full-field mechanical measurements, directional calibration can require many loading directions, and selecting an appropriate analytical form is nontrivial. This study proposes a physics-informed framework for discovering yield functions from full-field displacement data and reaction force data, without stress observations, plastic strain measurements, direct yield surface data, or a prescribed parametric yield function. The framework identifies the yield function as a mechanically constrained constitutive component inside elastoplastic stress integration, rather than through direct stress-space supervision. The yield function is represented by a convex neural network that enforces convexity and positive homogeneity of degree one while imposing the assumed tension-compression symmetry, and this neural yield function is trained with a differentiable stress update and a physics-informed force equilibrium loss across multiple loading cases. The proposed framework is validated using finite element (FE) benchmark studies with von Mises, Hill 1948, and Yld2000-2d yield functions, assessing yield contour agreement, displacement-noise sensitivity, identifiability through plastically active stress states, epistemic uncertainty, and polynomial-surrogate deployment. This study provides a mechanics-constrained pathway for discovering anisotropic yield functions from displacement and force data while keeping the identified component within the structure of elastoplastic stress integration.

Preserving Plasticity in Continual Learning via Dynamical Isometry

Continual training of deep neural networks under non-stationarity often leads to a progressive loss of plasticity, eventually limiting further learning. We relate plasticity to the empirical Neural Tangent Kernel, and identify dynamical isometry (the condition that layer-wise Jacobian singular values remain close to one) as a key mechanism for preserving plasticity in continual learning. We revisit a class of networks that are almost-everywhere isometric while remaining universal Lipschitz function approximators, demonstrating that near-dynamical isometry is compatible with expressive nonlinear representations. For general architectures, we propose an efficient isometry-promoting regularization scheme and identify a novel mechanism by which it can reactivate dormant ReLU units. Building on this, we introduce AdamO, an Adam-style adaptive optimizer that decouples isometry regularization from gradient updates, analogous to AdamW. We further reinterpret prior plasticity-preserving approaches through the lens of dynamical isometry, showing that they target only a partial measure of isometry. Across supervised and reinforcement-learning continual-learning benchmarks designed to induce plasticity loss, our methods consistently match or outperform existing approaches.