Computational Modeling in Science

🚀 Computational Physics: The Power of Science and Computing Computational Physics integrates physics, applied mathematics, and computer science to solve complex problems through numerical simulations. For professionals in modeling, machine learning, or data science, mastering this field unlocks groundbreaking applications. --- 🔍 What is Computational Physics? Many physical problems lack exact solutions, requiring numerical methods to obtain approximate answers. Computational Physics enables us to: ✔ Solve differential equations in complex systems. ✔ Simulate phenomena that can't be tested in labs. ✔ Model chaotic systems like climate and turbulence. ✔ Process and interpret vast scientific data. --- ⚙ Key Methods and Techniques 📌 Finite Difference (FDM), Finite Element (FEM), and Finite Volume (FVM) – Essential for engineering, fluid dynamics, and electromagnetism. 📌 Molecular Dynamics (MD) & Monte Carlo (MC) – Used in biophysics and materials science. 📌 Numerical Linear Algebra – LU decomposition, FFT, and conjugate gradient methods for efficient computations. --- 🌍 Applications of Computational Physics 🔭 Astrophysics & Cosmology – Simulating black holes, galaxy evolution and finding new exoplanets. ⚛ Particle Physics & Quantum Mechanics – Modeling high-energy collisions. 🌪 Computational Fluid Dynamics (CFD) – Applied in weather forecasting and aerodynamics. 🔬 Materials Science – Simulating semiconductors, nanotechnology, and superconductors. 🤖 Machine Learning & Physics – Neural Networks for solving PDEs and accelerating simulations. --- 🛠 Essential Tools & Programming Languages 💻 Languages – Python (NumPy, SciPy), C/C++, Fortran, Julia. 📊 Software & Frameworks – MATLAB, COMSOL, OpenFOAM, LAMMPS, GROMACS, ROOT. --- 🚀 Future Trends & Challenges ⚡ High-Performance Computing (HPC) – Enabling precise simulations. 🧠 AI in Physics – Implementing Physics-Informed Neural Networks (PINNs). 💡 Quantum Computing – Potential breakthroughs in quantum mechanics. --- 🔗 Conclusion From exploring the universe to predicting climate patterns and designing materials, Computational Physics is a powerful tool shaping science and technology. Mastering scientific programming and numerical modeling is an excellent starting point. 💬 Have you worked with Computational Physics before? Or interested in learning more? Let’s discuss in the comments! 👇

Learning molecular dynamics across timescales with generative models Molecular dynamics (MD) simulations track atoms one by one, step by step, revealing at atomic resolution how molecules fold, bind, and change shape. But there is a fundamental bottleneck: to remain numerically stable, MD must advance in femtosecond increments—even when the processes of real interest, like protein folding or drug unbinding, unfold over microseconds to seconds. Bridging this timescale gap is one of the central challenges of computational chemistry and biophysics. Juan Viguera Diez, Mathias Schreiner, and Simon Olsson address this with TITO (Transferable Implicit Transfer Operators), a deep generative framework that learns the statistical rules of molecular motion without ever taking a single femtosecond integration step. Instead of simulating trajectories step by step, TITO learns the transition probability distribution between molecular configurations at arbitrary lag times, using equivariant flow matching over a continuous normalizing flow. Trained jointly on small organic molecules and short peptides, the model internalizes both chemistry and timescale—and can then generate new trajectories at whatever time resolution is needed. The results are compelling. TITO faithfully reproduces Boltzmann equilibrium distributions and relaxation dynamics across hundreds of unseen molecules and peptides. More remarkably, it uncovers metastable conformational states that long conventional MD simulations miss—states later confirmed by replica exchange and ultralong trajectories. Trained only on tetrapeptides, it extrapolates qualitatively to peptides twice as large, guided by a simple physical prior from polymer scaling theory. On a single GPU, it achieves roughly 10 milliseconds of simulated physical time per day, versus a few microseconds for conventional MD—a speedup of four orders of magnitude. For drug discovery and materials science, this matters concretely. Conformational sampling—understanding which shapes a molecule can adopt—remains a computational bottleneck in hit identification and lead optimization. A transferable generative model that captures both thermodynamics and kinetics at a fraction of the cost of brute-force MD could meaningfully accelerate early-stage screening pipelines and reduce dependence on the most expensive compute infrastructure. Paper: Diez et al., Science Advances (2026) — CC BY-NC 4.0 | https://lnkd.in/eKBnKd4G #MachineLearning #DeepLearning #MolecularDynamics #ComputationalChemistry #GenerativeAI #DrugDiscovery #MaterialsScience #Biophysics #AIforScience #MLforScience #FlowMatching #GenerativeModels #Cheminformatics #ComputationalBiology

The paper below introduces a novel computational framework for understanding&modeling the interaction of molecules through the concept of “molecular holograms” I.e., spatiotemporal representations that encode the quantum&chemical properties of molecules (e.g., electronic distributions and reactive behaviors). The computational approach combines quantum mechanics, ML, & holographic imaging techniques to build a predictive&interpretable model of molecular systems. Molecular holography refers to a high-dimensional representation of molecules that encapsulates their spatial/temporal properties including electronic distributions, spin states, and other quantum mechanical descriptors. Spatiotemporal modeling involves tracking the dynamic behavior of molecules in space&time by integrating quantum mechanical simulations w/data-driven models that account for complex temporal dependencies, such as reaction kinetics. Methods: Time-dependent Density Functional Theory was used to simulate the electronic structure of molecules while molecular dynamics simulations provided insight into temporal evolution. The molecular holograms are generated by encoding wavefunction data into a multidimensional space using Fourier transforms integrating position, momentum, &electronic density. DL models (e.g., graph neural networks, recurrent networks) are trained on holographic data to learn patterns and predict outcomes like reactivity&stability. The molecular holograms enable precise predictions of reaction pathways, transition states, and activation energies. The method facilitates the design of molecules with desired properties by analyzing holograms for stability, reactivity, &functionality. The framework can identify molecular interactions in biological environments, aiding in drug-target binding predictions. The authors demonstrate the effectiveness of the method by applying it to a diverse dataset of molecular systems, including organic reactions, enzyme dynamics, & nanomaterial design. Comparative analysis shows that holographic models outperform traditional descriptors (e.g., molecular fingerprints) in terms of predictive accuracy&interpretability. This framework was able to predict complex non-linear phenomena (e.g., electron delocalization&excited-state dynamics). Molecular holograms provide a visually interpretable & mathematically rigorous framework; The integration of ML accelerates computations without compromising accuracy; The framework is applicable across a wide range of molecular systems. Unfortunately the computational cost remains high for large-scale systems & holographic encoding is sensitive to noise in input data, which may limit accuracy for certain classes of molecules. Future steps: Develop noise-robust holographic encoding algos; Scale up the approach for macromolecular systems (e.g., proteins, polymers); Extend the temporal resolution for ultra-fast processes (e.g., femtosecond reactions).

How do we bring AI to scientific modeling? The standard approach has been AI to augment existing numerical simulations. In a new work https://lnkd.in/gFMUvUbB we show this approach is fundamentally limited. In contrast, using the end-to-end AI approach of Neural Operators to completely replace numerical solvers helps overcome this limitation both in theory and in practice. Current augmentation approaches use AI as a closure model while keeping a coarse-grid numerical solver in the loop. We show that such approaches are generally unable to reach full fidelity, even if we make the closure models stochastic, providing them with history information and even unlimited ground-truth training data from full-fidelity solvers. This is because the closure model is forced to be at the same coarse resolution as the (cheap and approximate) numerical solver, and their combination does not result in high-fidelity solutions. In contrast, Neural Operators do not suffer from this limitation since they operate at any resolution and learn the mapping between functions. Neural Operators are first trained on coarse-grid approximate solvers, since we can generate lots of training data, and only use a small amount of expensive data from high-fidelity solvers in addition to physics-based losses to fine-tune the Neural Operator model for strong generalization. The key is that the Neural Operator model operates on any resolution, and can thus, accept data at multiple resolutions for training efficiently, without burdensome data-generation requirements.  Thus, Neural Operators fundamentally change how we apply AI to scientific domains.

🚀 DD-FEM: Train Small, Model Big (Local Training → Global Assembly) What if we could learn physics locally—on small patches—then assemble those learned “data-driven elements” to solve much larger PDE problems without retraining? That’s the idea behind DD-FEM (Data-Driven Finite Element Method): ✅ Locally trained, data-driven basis functions ✅ Finite-element-style local-to-global assembly ✅ Governing equations still enforced (not a black box) Why it matters (results we’re excited about): + >1000× speedup with <1% relative error on lattice-type elasticity—showing globally accurate solutions assembled from small, locally trained components. + 23.7× speedup with <4% error for scaled-up steady Navier–Stokes porous-media flow using DG-style coupling.  + 662× speedup with ~1% error for time-dependent Burgers dynamics, while generalizing across space/time from local training.  + A single learned manifold can represent both Poisson and Burgers, trained on local 2×2 subdomains (4,000 Poisson + 101,000 Burgers snapshots). The big picture: DD-FEM keeps the numerical-method rigor and modularity we trust—while gaining the reuse and scalability we want from data-driven models. 📄 If you’re curious about the framework and results, see our paper “Defining Foundation Models for Computational Science: A Call for Clarity and Rigor.” + Link to the paper: https://lnkd.in/gWSHPAqj #SciML #Numerical #Methods #Finite #Elements #Model #Reduction #Domain #Decomposition #HPC #PDE #Foundation #Models #libROM

Biomechanical Simulation: A Pure CAE Perspective How close can our simulations get to reality when the system itself is biologically complex? Biomechanical simulation is not merely about meshing geometry or running a solver. It is about capturing highly nonlinear, anisotropic, and time-dependent behavior within a numerically stable and physically consistent framework. From a computational standpoint, these models typically integrate: ✔ Finite Element Analysis (FEA) for soft tissues and structural response ✔ Computational Fluid Dynamics (CFD) for airflow and hemodynamics ✔ Fluid–Structure Interaction (FSI) for coupled fluid–tissue mechanics ✔ Multibody dynamics for kinematic systems ✔ Advanced constitutive laws (hyperelasticity, viscoelasticity, anisotropy) The real challenge is not model construction, it's model credibility: Strong variability in biological material properties Uncertain and idealized boundary conditions Large deformation, contact, and nonlinear convergence issues Limited experimental datasets for validation High numerical sensitivity to parameters and mesh density Unlike conventional mechanical components, biological systems do not come with standardized material datasheets. Correlation, parameter identification, and stability control become critical steps in the simulation workflow. The GIF from Oklahoma State University beautifully illustrates how numerical modeling can reveal transient airflow patterns in an elastic lung model, phenomena that are impossible to observe directly in vivo: https://lnkd.in/d_b3DTvT Biomechanical simulation is where advanced computational mechanics meet the complexity of life itself. #BiomechanicalSimulation #CAE #FiniteElementAnalysis #CFD #FSI #NonlinearAnalysis #ComputationalMechanics #EngineeringSimulation

🚀 Our perspective is out in Nature Portfolio! We present a roadmap for Multimodal Foundation Models (MFMs) — large AI models pretrained across multi-omics and multi-timepoint data — to serve as the computational backbone for building virtual cells. Read the full paper in Nature: https://lnkd.in/g3yzvA_f 🔎 Why MFMs? Biology is inherently multimodal, and molecular layers are deeply interconnected and context-specific. MFMs aims to integrate these layers to uncover shared biological principles that govern diverse cell states, offering a unified substrate for downstream inference. 🧠 What’s new? 💡 From hypothesis-driven to data-centric workflows: MFMs shift biology’s paradigm. Instead of crafting bespoke models for narrow tasks, we can now pretrain over massive datasets, distill foundational knowledge, and refine insights through lab-in-the-loop experimentation—where models guide experiments, and experiments update models. 🧬 Conditional gene regulation: MFMs go beyond static models. By training across multiple omics layers (e.g., chromatin accessibility, transcriptomics), they can learn context-specific gene functions and regulatory programs—key to understanding development and disease. 🧪 In silico perturbation: Biology’s combinatorial complexity is immense—thousands of genes, millions of interactions. MFMs provide a framework to simulate perturbations before wet-lab execution. Trained on CRISPR perturb-seq data, they can predict molecular responses across cell types, tissues, and time—enabling programmable biology at scale. ⚙️ What makes MFMs possible? Envisioned techniques include: - Unified tokenization from nucleotides to pathways - Hybrid attention across intra- and inter-modal interactions - Prompt-driven multitasking for temporal prediction, conditional generation, and modality translation - Human knowledge integration from curated databases and biomedical literature These design principles translate the architecture of foundation models into the molecular domain. ⚠️ What are the challenges? MFMs aren’t just about scale—they demand accessibility, reliability, and transparency. - Low-resource learning techniques (e.g., LoRA, adapters) are vital for democratizing training - Human-agnostic benchmarks are needed, as conventional labels may punish models that uncover novel biology - Uncertainty modeling is essential to mitigate hallucinations and increase scientific trust Interpretability and ethical stewardship must be foundational in this emerging ecosystem. Kudos to all co-authors for the collective effort and vision: Haotian Cui Alejandro Tejada Lapuerta #MariaBrbic Julio Saez Rodriguez Simona Cristea Hani Goodarzi Mo Lotfollahi Fabian Theis Let’s build the future of virtual cells together.

Physics-Informed Neural Networks (PINNs) are a powerful way to combine physical laws with machine learning. By adding physics into the model-building process, we can create more accurate and reliable models that not only learn from data but also use scientific knowledge we already have. Here's how they work: ➡️ The Balance of Theory and Experiment Science usually relies on two things: - Theory: A set of principles that explains how something works. - Experiment: Testing those principles to refine the theory. But with the rise of machine learning (ML) and a huge amount of data, data-driven methods are becoming popular. They allow us to solve scientific problems without needing a theory first. ➡️ Predicting New Results A common goal in science is to predict new outcomes based on the data we already have. Traditional neural networks are used to do this by learning from data points and adjusting to reduce errors between predictions and actual data. ➡️ Limitations of Data-Driven Models However, relying only on data can be tricky. A network might fit the data well within its range, but it often struggles to predict anything outside that range. This shows the limits of a purely data-driven approach—it doesn’t always truly "understand" the problem. ➡️ How PINNs Solve This PINNs solve this issue by integrating physical laws with data learning. Using our knowledge of the physical world—like differential equations—PINNs ensure that the model sticks to known science. ➡️ A Real-World Example For example, with a damped harmonic oscillator (a common physics problem), the network’s loss function combines: - The error between predicted and actual data. - The residuals from the physical equations. This allows the network to generalize better and make predictions beyond the experimental data, while still following the rules of physics.

A common feature in #materials and #chemicals modelling is the presence of activity cliffs: regions where a small change in input leads to a large change in output. These cliffs can confuse models and mislead decision-making. 📉 If you've ever tried to model a discontinuity with a Gaussian Process, you'll have seen they struggle with sudden changes. This is because GPs assume the function is smooth, so can't accurately fit the jump. Importantly, their uncertainty is also flat across the whole response: the GP doesn't “know��� it is inaccurate near the cliff. 📉 Traditional statistical models, like Chebyshev polynomial modelling, also struggle with activity cliffs, "ringing" near the jump: this is the Gibbs phenomenon (https://lnkd.in/eqjcRM4J) and is a fundamental limitation of using smooth functions to model sharp discontinuities. 📈 Methods like Random Forests and #Alchemite that don’t assume smoothness avoid these issues. They capture the jump more accurately, and their uncertainty increases near the discontinuity, realistically reflecting the limits of what the model knows. When using #MachineLearning to model complex data, it’s critical to choose methods that match the behaviour you expect to see: understanding the strengths and limitations of each approach helps ensure models are both accurate and trustworthy

One of the most fascinating aspects of engineering is how a physical phenomenon can be translated into mathematical equations, structured into matrix systems, and solved to uncover the unknowns that define a problem often represented at the nodes of a numerical model. In real life, most engineering problems are inherently complex. To understand them, predict their behavior, and make sound decisions, engineers rely on models: simplified yet powerful representations of reality that capture the essential physics of a system. These models allow us to analyze responses, optimize designs, comply with regulations, and guide strategic technical decisions. While physical modeling through laboratory or field experiments remains valuable, it is often time-consuming, costly, and limited. This is why mathematical and numerical modeling, supported by modern computational power, has become indispensable across all branches of engineering. Techniques such as the Finite Element Method (FEM) enable us to discretize complex domains, apply governing equations, and solve large systems efficiently and accurately. From problem definition and simplification, to model validation, sensitivity analysis, and interpretation of results, numerical modeling is not just about computation : it is about engineering judgment, scientific rigor, and deep domain expertise. This is where engineering becomes both science and art: transforming complexity into clarity, and equations into solutions that shape the real world. 🔬📐🌍 #Engineering #GeotechnicalEngineering #FiniteElementMethod #NumericalModeling #ScientificComputing #AppliedMathematics #STEM #CivilEngineering #ComputationalEngineering #EngineeringExcellence