PRISM-UDE: Physics-Regularized Iterative Symbolic Modeling of 3nm FinFETs via Universal Differential Equation

Neural network (NN)-based transistor models have proven to be a promising solution to accelerate device modeling. Although these models demonstrated remarkable speedup in circuit simulations, they were not rigorously tested in complex tasks within standard EDA tool flows for chip design, which require very high transistor model precision to provide accurate results and converge properly. To investigate their full potential and promise, we develop NN-based transistor models that accurately predict the drain current and various charges for a 3nm nanosheet, and employ the models for a standard cell library characterization. This enables us to evaluate the accuracy and speedup under a very wide range of SPICE simulations, benchmarking our models against an industry-standard implementation. Since standard cell library characterization involves thousands of SPICE simulations, it serves as a great benchmark for transistor model quality and consistency. To further evaluate model accuracy, we also perform analog- and digital-level simulations for various complex circuits. Our experiments reveal that NN-based models are a valid alternative to existing transistor compact models, capable of producing accurate delay and power estimates with a much higher speed and convergence time compared to the conventional Verilog-A-based industry-standard (BSIM-CMG) transistor compact model. Our NNs achieve sub-0.11% and 0.76% errors for delay and total power evaluation on a wide range of circuits, while taking up to 8 times less time during the standard cell library characterization. Additionally, we provide comprehensive information and guidance on NN accuracy, unveiling the relationship between NN size and its effect on circuit simulation precision.

The drift-diffusion equations are the fundamental building block for the modeling, design, and optimization of modern semiconductor devices. Solutions of the drift-diffusion equations rarely give way to analytical forms, and exhibit strong nonlinearity, with the solutions ranging over several orders of magnitude. These attributes have led to research into numerical solvers, and while many commercial Technology Computer Aided Design (TCAD) packages now exist, they may require long simulation times, exhibit sensitivity to initial guesses, and rely on mesh generation. Further, these solvers are not well suited for inverse problems where the design parameters of a device are retrieved from its desired functional characteristics. To address all these challenges, this work demonstrates the usage of Physics Informed Neural Networks (PINNs) for the solution of the drift-diffusion equations with errors comparable to traditional simulation software using advanced training techniques to solve problems beyond the well-behaved equilibrium regime. For this study, we specifically choose an abrupt PN-junction in both one and two dimensions that features extreme gradients of the carrier concentrations at the Ohmic contacts and an abrupt discontinuity in the doping profile. Using the proposed PINN architecture, we are able to retrieve accurate results for all observables under all bias conditions. Additionally, we apply the PINN solver towards the inverse problem of retrieving doping profiles directly from desired electric field distributions or electrostatic potentials under non-zero applied biases. Our work establishes a powerful PINNs-based approach for semiconductor device modeling with the ability to solve inverse problems under non-equilibrium conditions for microelectronics research and discovery.

What mathematical functions do neural network components learn? Symbolic distillation addresses this question by expressing neural network components with interpretable, closed-form mathematical expressions that expose the functional structure learned during training. We develop symbolic distillation as a systematic, architecture-agnostic methodology, and release our approach as the open-source SymTorch package - a PySR-powered library built natively for the PyTorch ecosystem. Applying this methodology across diverse architectures, we find that SymTorch is successful in the automated discovery of physical laws. Specifically, our approach (1) recovers pairwise interaction forces from graph neural networks trained on empirical $n$-body observations, (2) distills the exact closed-form PDE/ODE solutions of multiple physical systems, including the value of constants, from physics-informed neural networks trained on sparse data, and (3) uncovers the chaotic dynamics of the Lorenz system from high-dimensional data, ultimately outperforming the base neural network on downstream prediction tasks. We further demonstrate the utility of our framework for model interpretability by providing an optimized implementation of SLIME - a symbolic extension to the LIME explainability method. SLIME consistently outperforms LIME across predictive metrics across eight popular classification and regression benchmarks, while still providing an interpretable local symbolic model. Lastly, we investigate replacing transformer MLP layers with symbolic surrogates: replacing 1-7 layers with symbolic approximations yields 2-19% throughput improvements and up to 18.7% VRAM reduction, with the resulting hybrid models lying on the Pareto front of throughput versus perplexity among open-source LLMs of comparable scale.