Dibakar Roy Sarkar

Adam J. Thorpe, Stepan Tretiakov, Dibakar Roy Sarkar et al.

Inverse problems challenge existing neural operator architectures because ill-posed inverse maps violate continuity, uniqueness, and stability assumptions. We introduce B2B${}^{-1}$, an inverse basis-to-basis neural operator framework that addresses this limitation. Our key innovation is to decouple function representation from the inverse map. We learn neural basis functions for the input and output spaces, then train inverse models that operate on the resulting coefficient space. This structure allows us to learn deterministic, invertible, and probabilistic models within a single framework, and to choose models based on the degree of ill-posedness. We evaluate our approach on six inverse PDE benchmarks, including two novel datasets, and compare against existing invertible neural operator baselines. We learn probabilistic models that capture uncertainty and input variability, and remain robust to measurement noise due to implicit denoising in the coefficient calculation. Our results show consistent re-simulation performance across varying levels of ill-posedness. By separating representation from inversion, our framework enables scalable surrogate models for inverse problems that generalize across instances, domains, and degrees of ill-posedness.

Pietro Zanotta, Dibakar Roy Sarkar, Honghui Zheng et al.

Controlling partial differential equations (PDEs) with learning-based policies remains fundamentally limited by fixed-dimensional representations: policies trained for a specific sensor, actuator, or agent configuration typically fail when the configuration changes. This limitation is particularly severe in multi-agent PDE control, where policies do not scale across population sizes without retraining. We address this challenge by introducing Cardinality Invariant Neural Operator Control (CINOC), reformulating PDE control as an operator learning problem that maps state fields to continuous control functions and trains them end-to-end through differentiable PDE solvers, yielding policies that naturally adapt to varying sensor and actuator configurations. Remarkably, CINOC policies trained on small swarms exhibit cardinality invariance, allowing for zero-shot transfer to significantly larger populations as well as robustness to partial agent failure. This scalability arises from agents sharing a common policy and coordinating through their physical environment, which produces an emergent self-normalization effect. To explain this phenomenon, we provide a theorem grounded in mean-field theory demonstrating that policy gradients computed from finite-agent systems converge to those of a continuous control limit. Empirically, we validate CINOC on tracking, stabilization, and density transport across linear, nonlinear, chaotic, and turbulent PDEs.

Dibakar Roy Sarkar, Ján Drgoňa, Somdatta Goswami

We present a data-driven control framework for partial differential equations (PDEs). Our approach integrates Time-Integrated Deep Operator Networks (TI-DeepONets) as differentiable PDE surrogate models within the Differentiable Predictive Control (DPC)-a self-supervised learning framework for constrained neural control policies. The TI-DeepONet architecture learns temporal derivatives and couples them with numerical integrators, while the DPC algorithm uses automatic differentiation to compute policy gradients by backpropagating the expectations of the optimal control loss through the learned TI-DeepONet. This approach enables efficient offline optimization of neural policies without the need for online optimization or supervisory controllers. We empirically demonstrate the proposed method across diverse PDE systems, including the heat, the nonlinear Burgers', and the reaction-diffusion equations. The learned policies achieve target tracking, constraint satisfaction, and curvature minimization objectives, while generalizing across distributions of initial conditions and parameters. Moreover, we demonstrate four orders of magnitude acceleration at inference time compared to nonlinear model predictive control benchmarks. These results highlight the promise of operator learning for scalable model-based control of PDEs.

Yue Luo, Dibakar Roy Sarkar, Rachel Herring Sangree et al.

Developing effective, domain-specific educational support systems is central to advancing AI in education. Although large language models (LLMs) demonstrate remarkable capabilities, they face significant limitations in specialized educational applications, including hallucinations, limited knowledge updates, and lack of domain expertise. Fine-tuning requires complete model retraining, creating substantial computational overhead, while general-purpose LLMs often provide inaccurate responses in specialized contexts due to reliance on generalized training data. To address this, we propose ARIA (Adaptive Retrieval Intelligence Assistant), a Retrieval-Augmented Generation (RAG) framework for creating intelligent teaching assistants across university-level courses. ARIA leverages a multimodal content extraction pipeline combining Docling for structured document analysis, Nougat for mathematical formula recognition, and GPT-4 Vision API for diagram interpretation, with the e5-large-v2 embedding model for high semantic performance and low latency. This enables accurate processing of complex educational materials while maintaining pedagogical consistency through engineered prompts and response controls. We evaluate ARIA using lecture material from Statics and Mechanics of Materials, a sophomore-level civil engineering course at Johns Hopkins University, benchmarking against ChatGPT-5. Results demonstrate 97.5% accuracy in domain-specific question filtering and superior pedagogical performance. ARIA correctly answered all 20 relevant course questions while rejecting 58 of 60 non-relevant queries, achieving 90.9% precision, 100% recall, and 4.89/5.0 average response quality. These findings demonstrate that ARIA's course-agnostic architecture represents a scalable framework for domain-specific educational AI deployment.

Vijay Kag, Dibakar Roy Sarkar, Birupaksha Pal et al.

Big data is transforming scientific progress by enabling the discovery of novel models, enhancing existing frameworks, and facilitating precise uncertainty quantification, while advancements in scientific machine learning complement this by providing powerful tools to solve inverse problems to identify the complex systems where traditional methods falter due to sparse or noisy data. We introduce two innovative neural operator frameworks tailored for discovering hidden physics and identifying unknown system parameters from sparse measurements. The first framework integrates a popular neural operator, DeepONet, and a physics-informed neural network to capture the relationship between sparse data and the underlying physics, enabling the accurate discovery of a family of governing equations. The second framework focuses on system parameter identification, leveraging a DeepONet pre-trained on sparse sensor measurements to initialize a physics-constrained inverse model. Both frameworks excel in handling limited data and preserving physical consistency. Benchmarking on the Burgers' equation and reaction-diffusion system demonstrates state-of-the-art performance, achieving average $L_2$ errors of $\mathcal{O}(10^{-2})$ for hidden physics discovery and absolute errors of $\mathcal{O}(10^{-3})$ for parameter identification. These results underscore the frameworks' robustness, efficiency, and potential for solving complex scientific problems with minimal observational data.

Anusha Agarwal, Dibakar Roy Sarkar, Somdatta Goswami

Traumatic brain injury (TBI) remains a major public health concern, with over 69 million cases annually worldwide. Finite element (FE) models offer high-fidelity predictions of brain deformation but are computationally expensive, requiring hours per simulation and limiting their clinical utility for rapid decision-making. This study benchmarks state-of-the-art neural operator (NO) architectures for rapid, patient-specific prediction of brain displacement fields, aiming to enable real-time TBI modeling in clinical and translational settings. We formulated TBI modeling as an operator learning problem, mapping subject-specific anatomical MRI, magnetic resonance elastography (MRE) stiffness maps, and demographic features to full-field 3D brain displacement predictions. Four architectures - Fourier Neural Operator (FNO), Factorized FNO (F-FNO), Multi-Grid FNO (MG-FNO), and Deep Operator Network (DeepONet) were trained and evaluated on 249 MRE datasets across physiologically relevant frequencies (20 - 90 Hz). MG-FNO achieved the highest accuracy (MSE = 0.0023, 94.3\% spatial fidelity) and preserved fine-scale features, while F-FNO converged 2$\times$ faster than standard FNO. DeepONet offered the fastest inference (14.5 iterations/s) with a 7$\times$ computational speed-up over MG-FNO, suggesting utility for embedded or edge computing applications. All NOs reduced computation time from hours to milliseconds without sacrificing anatomical realism. NOs provide an efficient, resolution-invariant approach for predicting brain deformation, opening the door to real-time, patient-specific TBI risk assessment, clinical triage support, and optimization of protective equipment. These results highlight the potential for NO-based digital twins of the human brain, enabling scalable, on-demand biomechanical modeling in both clinical and population health contexts.