Sourav Saha

Reza T. Batley, Sourav Saha

Inverse design of heterogeneous material microstructures is a fundamentally ill-posed and famously computationally expensive problem. This is exacerbated by the high-dimensional design spaces associated with finely resolved images, multimodal input property streams, and a highly nonlinear forward physics. Whilst modern generative models excel at accurately modeling such complex forward behavior, most of them are not intrinsically structured to support fast, stable \emph{deterministic} inversion with a physics-informed bias. This work introduces Janus, a unified generative-predictive framework to address this problem. Janus couples a deep encoder-decoder architecture with a predictive KHRONOS head, a separable neural architecture. Topologically speaking, Janus learns a latent manifold simultaneously isometric for generative inversion and pruned for physical prediction; the joint objective inducing \emph{disentanglement} of the latent space. Janus is first validated on the MNIST dataset, demonstrating high-fidelity reconstruction, accurate classification and diverse generative inversion of all ten target classes. It is then applied to the inverse design of heterogeneous microstructures labeled with thermal conductivity. It achieves a forward prediction accuracy $R^2=0.98$ (2\% relative error) and sub-5\% pixelwise reconstruction error. Inverse solutions satisfy target properties to within $1\%$ relative error. Inverting a sweep through properties reveal smooth traversal of the latent manifold, and UMAP visualization confirms the emergence of a low-dimensional, disentangled manifold. By unifying prediction and generation within a single latent space, Janus enables real-time, physics-informed inverse microstructure generation at a lower computational cost typically associated with classical optimization-based approaches.

Chanwook Park, Sourav Saha, Jiachen Guo et al.

Artificial intelligence (AI) has revolutionized software development, shifting from task-specific codes (Software 1.0) to neural network-based approaches (Software 2.0). However, applying this transition in engineering software presents challenges, including low surrogate model accuracy, the curse of dimensionality in inverse design, and rising complexity in physical simulations. We introduce an interpolating neural network (INN), grounded in interpolation theory and tensor decomposition, to realize Engineering Software 2.0 by advancing data training, partial differential equation solving, and parameter calibration. INN offers orders of magnitude fewer trainable/solvable parameters for comparable model accuracy than traditional multi-layer perceptron (MLP) or physics-informed neural networks (PINN). Demonstrated in metal additive manufacturing, INN rapidly constructs an accurate surrogate model of Laser Powder Bed Fusion (L-PBF) heat transfer simulation, achieving sub-10-micrometer resolution for a 10 mm path in under 15 minutes on a single GPU. This makes a transformative step forward across all domains essential to engineering software.

Reza T. Batley, Sourav Saha

Training - the optimisation of complex models - is traditionally performed through small, local, iterative updates [D. E. Rumelhart, G. E. Hinton, R. J. Williams, Nature 323, 533-536 (1986)]. Approximating solutions through truncated gradients is a paradigm dating back to Cauchy [A.-L. Cauchy, Comptes Rendus Mathématique 25, 536-538 (1847)] and Newton [I. Newton, The Method of Fluxions and Infinite Series (Henry Woodfall, London, 1736)]. This work introduces the Method of Infinite Descent, a semi-analytic optimisation paradigm that reformulates training as the direct solution to the first-order optimality condition. By analytical resummation of its Taylor expansion, this method yields an exact, algebraic equation for the update step. Realisation of the infinite Taylor tower's cascading resummation is formally derived, and an exploitative algorithm for the direct solve step is proposed. This principle is demonstrated with the herein-introduced AION (Analytic, Infinitely-Optimisable Network) architecture. AION is a model designed expressly to satisfy the algebraic closure required by Infinite Descent. In a simple test problem, AION reaches the optimum in a single descent step. Together, this optimiser-model pair exemplify how analytic structure enables exact, non-iterative convergence. Infinite Descent extends beyond this example, applying to any appropriately closed architecture. This suggests a new class of semi-analytically optimisable models: the \emph{Infinity Class}; sufficient conditions for class membership are discussed. This offers a pathway toward non-iterative learning.

Reza T. Batley, Chanwook Park, Wing Kam Liu et al.

Data-driven science and computation have advanced immensely to construct complex functional relationships using trainable parameters. However, efficiently discovering interpretable and accurate closed-form expressions from complex dataset remains a challenge. The article presents a novel approach called Explainable Hierarchical Deep Learning Neural Networks or Ex-HiDeNN that uses an accurate, frugal, fast, separable, and scalable neural architecture with symbolic regression to discover closed-form expressions from limited observation. The article presents the two-step Ex-HiDeNN algorithm with a separability checker embedded in it. The accuracy and efficiency of Ex-HiDeNN are tested on several benchmark problems, including discerning a dynamical system from data, and the outcomes are reported. Ex-HiDeNN generally shows outstanding approximation capability in these benchmarks, producing orders of magnitude smaller errors compared to reference data and traditional symbolic regression. Later, Ex-HiDeNN is applied to three engineering applications: a) discovering a closed-form fatigue equation, b) identification of hardness from micro-indentation test data, and c) discovering the expression for the yield surface with data. In every case, Ex-HiDeNN outperformed the reference methods used in the literature. The proposed method is built upon the foundation and published works of the authors on Hierarchical Deep Learning Neural Network (HiDeNN) and Convolutional HiDeNN. The article also provides a clear idea about the current limitations and future extensions of Ex-HiDeNN.

Reza T. Batley, Sourav Saha

Contemporary models of high dimensional physical systems are constrained by the curse of dimensionality and a reliance on dense data. We introduce KHRONOS (Kernel Expansion Hierarchy for Reduced Order, Neural Optimized Surrogates), an AI framework for model based, model free and model inversion tasks. KHRONOS constructs continuously differentiable target fields with a hierarchical composition of per-dimension kernel expansions, which are tensorized into modes and then superposed. We evaluate KHRONOS on a canonical 2D, Poisson equation benchmark: across 16 to 512 degrees of freedom (DoFs), it obtained L_2-square errors of 5e-4 down to 6e-11. This represents a greater than 100-fold gain over Kolmogorov Arnold Networks (which itself reports a 100 times improvement on MLPs/PINNs with 100 times fewer parameters) when controlling for the number of parameters. This also represents a 1e6-fold improvement in L_2-square error compared to standard linear FEM at comparable DoFs. Inference complexity is dominated by inner products, yielding sub-millisecond full-field predictions that scale to an arbitrary resolution. For inverse problems, KHRONOS facilitates rapid, iterative level set recovery in only a few forward evaluations, with sub-microsecond per sample latency. KHRONOS's scalability, expressivity, and interpretability open new avenues in constrained edge computing, online control, computer vision, and beyond.