Root-cause analysis in controlled time dependent systems poses a major challenge in applications. Especially energy systems are difficult to handle as they exhibit instantaneous as well as delayed effects and if equipped with storage, do have a memory. In this paper we adapt the causal root-cause analysis method of Budhathoki et al. [2022] to general time-dependent systems, as it can be regarded as a strictly causal definition of the term "root-cause". Particularly, we discuss two truncation approaches to handle the infinite dependency graphs present in time-dependent systems. While one leaves the causal mechanisms intact, the other approximates the mechanisms at the start nodes. The effectiveness of the different approaches is benchmarked using a challenging data generation process inspired by a problem in factory energy management: the avoidance of peaks in the power consumption. We show that given enough lags our extension is able to localize the root-causes in the feature and time domain. Further the effect of mechanism approximation is discussed.

Standard approaches for uncertainty quantification in deep learning and physics-informed learning have persistent limitations. Indicatively, strong assumptions regarding the data likelihood are required, the performance highly depends on the selection of priors, and the posterior can be sampled only approximately, which leads to poor approximations because of the associated computational cost.This paper introduces and studies confidence interval (CI) estimation for deterministic partial differential equations as a novel problem.That is, to propagate confidence, in the form of CIs, from data locations to the entire domain with probabilistic guarantees.We propose a method, termed Physics-Informed Confidence Propagation (PICProp), based on bi-level optimization to compute a valid CI without making heavy assumptions.We provide a theorem regarding the validity of our method, and computational experiments, where the focus is on physics-informed learning.

Physical models in the form of partial differential equations serve as important priors for many under-constrained problems. One such application is tumor treatment planning, which relies on accurately estimating the spatial distribution of tumor cells within a patient's anatomy. While medical imaging can detect the bulk of a tumor, it cannot capture the full extent of its spread, as low-concentration tumor cells often remain undetectable, particularly in glioblastoma, the most common primary brain tumor. Machine learning approaches struggle to estimate the complete tumor cell distribution due to a lack of appropriate training data. Consequently, most existing methods rely on physics-based simulations to generate anatomically and physiologically plausible estimations. However, these approaches face challenges with complex and unknown initial conditions and are constrained by overly rigid physical models. In this work, we introduce a novel method that integrates data-driven and physics-based cost functions, akin to Physics-Informed Neural Networks (PINNs).

Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. The typical approach is to incorporate physical domain knowledge as soft constraints on an empirical loss function and use existing machine learning methodologies to train the model. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena for even slightly more complex problems. In particular, we analyze several distinct situations of widespread physical interest, including learning differential equations with convection, reaction, and diffusion operators. We provide evidence that the soft regularization in PINNs, which involves PDE-based differential operators, can introduce a number of subtle problems, including making the problem more ill-conditioned.

As a pivotal component to attaining generalizable solutions in human intelligence, reasoning provides great potential for reinforcement learning (RL) agents' generalization towards varied goals by summarizing part-to-whole arguments and discovering cause-and-effect relations. However, how to discover and represent causalities remains a huge gap that hinders the development of causal RL. In this paper, we augment Goal-Conditioned RL (GCRL) with Causal Graph (CG), a structure built upon the relation between objects and events. We novelly formulate the GCRL problem into variational likelihood maximization with CG as latent variables. To optimize the derived objective, we propose a framework with theoretical performance guarantees that alternates between two steps: using interventional data to estimate the posterior of CG; using CG to learn generalizable models and interpretable policies. Due to the lack of public benchmarks that verify generalization capability under reasoning, we design nine tasks and then empirically show the effectiveness of the proposed method against five baselines on these tasks. Further theoretical analysis shows that our performance improvement is attributed to the virtuous cycle of causal discovery, transition modeling, and policy training, which aligns with the experimental evidence in extensive ablation studies.

Large Language Models (LLMs) have shown state-of-the-art performance in a variety of tasks, including arithmetic and reasoning; however, to gauge the intellectual capabilities of LLMs, causal reasoning has become a reliable proxy for validating a general understanding of the mechanics and intricacies of the world similar to humans. Previous works in natural language processing (NLP) have either focused on open-ended causal reasoning via causal commonsense reasoning (CCR) or framed a symbolic representation-based question answering for theoretically backed-up analysis via a causal inference engine. The former adds an advantage of real-world grounding but lacks theoretically backed-up analysis/validation, whereas the latter is far from real-world grounding. In this work, we bridge this gap by proposing the COLD (Causal reasOning in cLosed Daily activities) framework, which is built upon human understanding of daily real-world activities to reason about the causal nature of events. We show that the proposed framework facilitates the creation of enormous causal queries ( 9 million) and comes close to the mini-turing test, simulating causal reasoning to evaluate the understanding of a daily real-world task. We evaluate multiple LLMs on the created causal queries and find that causal reasoning is challenging even for activities trivial to humans. We further explore (the causal reasoning abilities of LLMs) using the backdoor criterion to determine the causal strength between events.

Deep Operator Networks (DeepONets) have become a central tool in data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk design based on fully connected layers acting on raw spatial or spatiotemporal coordinates struggles to represent sharp gradients, boundary layers, and non-periodic structures commonly found in PDEs posed on bounded domains with Dirichlet or Neumann boundary conditions. To address these limitations, we introduce the Spectral-Embedded DeepONet (SEDONet), a new DeepONet variant in which the trunk is driven by a fixed Chebyshev spectral dictionary rather than coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias tailored to bounded domains, enabling the learned operator to capture fine-scale non-periodic features that are difficult for Fourier or MLP trunks to represent. SEDONet is evaluated on a suite of PDE benchmarks including 2D Poisson, 1D Burgers, 1D advection-diffusion, Allen-Cahn dynamics, and the Lorenz-96 chaotic system, covering elliptic, parabolic, advective, and multiscale temporal phenomena, all of which can be viewed as canonical problems in computational mechanics. Across all datasets, SEDONet consistently achieves the lowest relative L2 errors among DeepONet, FEDONet, and SEDONet, with average improvements of about 30-40% over the baseline DeepONet and meaningful gains over Fourier-embedded variants on non-periodic geometries. Spectral analyses further show that SEDONet more accurately preserves high-frequency and boundary-localized features, demonstrating the value of Chebyshev embeddings in non-periodic operator learning. The proposed architecture offers a simple, parameter-neutral modification to DeepONets, delivering a robust and efficient spectral framework for surrogate modeling of PDEs on bounded domains.

Designing materials with controlled heat flow at the nano-scale is central to advances in microelectronics, thermoelectrics, and energy-conversion technologies. At these scales, phonon transport follows the Boltzmann Transport Equation (BTE), which captures non-diffusive (ballistic) effects but is too costly to solve repeatedly in inverse-design loops. Existing surrogate approaches trade speed for accuracy: fast macroscopic solvers can overestimate conductivities by hundreds of percent, while recent data-driven operator learners often require thousands of high-fidelity simulations. This creates a need for a fast, data-efficient surrogate that remains reliable across ballistic and diffusive regimes. We introduce a Physics-Enhanced Deep Surrogate (PEDS) that combines a differentiable Fourier solver with a neural generator and couples it with uncertainty-driven active learning. The Fourier solver acts as a physical inductive bias, while the network learns geometry-dependent corrections and a mixing coefficient that interpolates between macroscopic and nano-scale behavior. PEDS reduces training-data requirements by up to 70% compared with purely data-driven baselines, achieves roughly 5% fractional error with only 300 high-fidelity BTE simulations, and enables efficient design of porous geometries spanning 12-85 W m$^{-1}$ K$^{-1}$ with average design errors of 4%. The learned mixing parameter recovers the ballistic-diffusive transition and improves out of distribution robustness. These results show that embedding simple, differentiable low-fidelity physics can dramatically increase surrogate data-efficiency and interpretability, making repeated PDE-constrained optimization practical for nano-scale thermal-materials design.

As artificial intelligence emerges as a transformative enabler for fusion energy commercialization, fast and accurate solvers become increasingly critical. In magnetic confinement nuclear fusion, rapid and accurate solution of the Grad-Shafranov equation (GSE) is essential for real-time plasma control and analysis. Traditional numerical solvers achieve high precision but are computationally prohibitive, while data-driven surrogates infer quickly but fail to enforce physical laws and generalize poorly beyond training distributions. To address this challenge, we present a Physics-Informed Neural Operator (PINO) that directly learns the GSE solution operator, mapping shape parameters of last closed flux surface to equilibrium solutions for realistic nonlinear current profiles. Comprehensive benchmarking of five neural architectures identifies the novel Transformer-KAN (Kolmogorov-Arnold Network) Neural Operator (TKNO) as achieving highest accuracy (0.25% mean L2 relative error) under supervised training (only data-driven). However, all data-driven models exhibit large physics residuals, indicating poor physical consistency. Our unsupervised training can reduce the residuals by nearly four orders of magnitude through embedding physics-based loss terms without labeled data. Critically, semi-supervised learning--integrating sparse labeled data (100 interior points) with physics constraints--achieves optimal balance: 0.48% interpolation error and the most robust extrapolation performance (4.76% error, 8.9x degradation factor vs 39.8x for supervised models). Accelerated by TensorRT optimization, our models enable millisecond-level inference, establishing PINO as a promising pathway for next-generation fusion control systems.

Optimal experimental design is a classic topic in statistics, with many well-studied problems, applications, and solutions. The design problem we study is the placement of sensors to monitor spatiotemporal processes, explicitly accounting for the temporal dimension in our modeling and optimization. We observe that recent advancements in computational sciences often yield large datasets based on physics-based simulations, which are rarely leveraged in experimental design. We introduce a novel model-based sensor placement criterion, along with a highly-efficient optimization algorithm, which integrates physics-based simulations and Bayesian experimental design principles to identify sensor networks that "minimize information loss" from simulated data. Our technique relies on sparse variational inference and (separable) Gauss-Markov priors, and thus may adapt many techniques from Bayesian experimental design. We validate our method through a case study monitoring air temperature in Phoenix, Arizona, using state-of-the-art physics-based simulations. Our results show our framework to be superior to random or quasi-random sampling, particularly with a limited number of sensors. We conclude by discussing practical considerations and implications of our framework, including more complex modeling tools and real-world deployments.