The heat method using FEM
Neurips 2025, AI for Science Workshop
Abstract: Recent advances in Neural Fields have enabled powerful, discretization-invariant methods for learning neural operators that approximate solutions of Partial Differential Equations (PDEs) on general geometries. Building on these developments, we introduce enf2enf, an encoder–decoder methodology for predicting steady-state Partial Differential Equations with non-parameterized geometric variability, based on recently proposed Equivariant Neural Field architectures. In enf2enf, input geometries are encoded into latent point cloud embeddings that inherently preserve geometric grounding and capture local phenomena……. [shortened for brevity]
| Paper PDF | Code JAX | Code Torch |
Computers & Fluids, 106929
Abstract: This paper introduces a novel surrogate modeling framework for aerodynamic applications based on Neural Fields. The proposed approach, MARIO (Modulated Aerodynamic Resolution Invariant Operator), addresses non parametric geometric variability through an efficient shape encoding mechanism and exploits the discretization-invariant nature of Neural Fields. It enables training on significantly downsampled meshes, while maintaining consistent accuracy during full-resolution inference. These properties allow for efficient modeling of diverse flow conditions, while reducing computational cost and memory requirements compared to traditional CFD solvers and existing surrogate methods. The framework is validated on two complementary datasets that reflect industrial constraints. First, the AirfRANS dataset consists in a two-dimensional airfoil benchmark with non-parametric shape variations. Performance evaluation of MARIO on this case demonstrates an order of magnitude improvement in prediction accuracy over existing methods across velocity, pressure, and turbulent viscosity fields, while accurately capturing boundary layer phenomena and aerodynamic coefficients. Second, the NASA Common Research Model features three-dimensional pressure distributions on a full aircraft surface mesh, with parametric control surface deflections.
| Paper PDF | Code Torch |
Scientific Reports, Vol. 14(1), pp. 25496
Abstract: This paper presents a methodology to learn surrogate models of steady state fluid dynamics simulations on meshed domains, based on Implicit Neural Representations (INRs). The proposed models can be applied directly to unstructured domains for different flow conditions, handle non-parametric 3D geometric variations, and generalize to unseen shapes at test time… [shortened for brevity]
| Paper PDF | Code | Data |
Why optimization people love quadratic forms ?
Gradient-based optimization repeatedly encounters objectives like f(x) = xᵀAx
because they have beautiful properties.
The gradient is ∇f = 2Ax
when A is symmetric.
The Hessian is ∇²f = 2A.
Immediately,
Exactly the same ideas appear in mechanics:
In a specific design optimization field called Topology Optimization a quadratic form has to be minimized.
Changing the density of an element changes how much strain energy that element can carry.
The optimizer therefore decides:
Where should material be placed so that the total strain energy—and therefore the compliance—is minimized?
The design variables are the element densities:
ρ = (ρ₁, ρ₂, ..., ρₙ)ᵀ
The topology optimization problem is:
minimize C(ρ) = fᵀu = uᵀK(ρ)u
subject to K(ρ)u = f
Σₑ ρₑ vₑ ≤ V*
ρₘᵢₙ ≤ ρₑ ≤ 1
e = 1,...,n
where:
The compliance objective is the quadratic form:
C = uᵀ K u
or equivalently:
C = fᵀ u
because:
K u = f
Column vectors:
C = u' * K * u;
Equivalent scalar product:
C = dot(u, K*u);
For sparse FEM matrices:
C = u' * (K*u);
Using matrix multiplication:
C = u' * K * u
Recommended scalar form:
C = dot(u, K*u)
For sparse FEM:
C = dot(u, K*u)
with
K = sparse(K)
Using matrix multiplication:
C = u.T @ K @ u
Equivalent:
C = np.dot(u, K @ u)
For sparse FEM matrices:
C = u @ (K @ u)
with:
from scipy.sparse import csr_matrix
K = csr_matrix(K)
The global compliance can be decomposed into element contributions:
C = Σₑ uₑᵀ Kₑ uₑ
where e runs over all elements: e = 1, …, n
with
uₑ = element displacement vector
Kₑ = element stiffness matrix
The element stiffness is interpolated as:
Kₑ(ρₑ) = ρₑᵖ Kₑ⁰
where:
p = penalization exponent
Kₑ⁰ = stiffness of solid material
Therefore:
C = Σₑ ρₑᵖ uₑᵀ Kₑ⁰ uₑ
The SIMP derivative is:
∂C/∂ρₑ = -p ρₑᵖ⁻¹ uₑᵀ Kₑ⁰ uₑ
which is the quantity used by OC (Optimality Criteria) and MMA update schemes.
In practice we are using SIMP formulation
The Solid Isotropic Material with Penalization (SIMP) method introduces a
relationship between the design variable and the Young modulus of each
element.
The interpolation is:
Eₑ(xₑ) = Eₘᵢₙ + xₑᵖ (E₀ − Eₘᵢₙ)
where:
Eₑ(xₑ) = Young modulus of element e
E₀ = Young modulus of solid material
Eₘᵢₙ = small stiffness assigned to void regions
xₑ = density variable of element e
p = penalization exponent (usually p ≈ 3)
The purpose of the penalization term:
xₑᵖ
is to discourage intermediate densities:
0 < xₑ < 1
and drive the solution toward a black-and-white topology:
solid → xₑ = 1
void → xₑ ≈ 0
The global compliance can be decomposed into element contributions:
C(x) = Σₑ Eₑ(xₑ) uₑᵀ k₀ uₑ
where:
uₑ = displacement vector of element e
k₀ = reference element stiffness matrix
Eₑ(xₑ) = SIMP interpolated Young modulus
Using the SIMP interpolation:
C(x)
=
Σₑ [Eₘᵢₙ + xₑᵖ(E₀ − Eₘᵢₙ)]
uₑᵀ k₀ uₑ
The sum notation means:
Σₑ = sum over all finite elements
e = 1, ..., N
The derivative used by optimization algorithms
(Optimality Criteria, MMA, etc.) is:
∂C/∂xₑ
=
-p xₑᵖ⁻¹ (E₀ − Eₘᵢₙ)
uₑᵀ k₀ uₑ
This sensitivity tells how the compliance changes when the material
density of element e is modified.
Summary: A hands-on, deeply technical tutorial on topology and structural optimization. The post breaks down the mathematics behind minimizing elastic potential energy (compliance) across a 2D grid of springs. It covers crucial implementation steps, including computing sensitivities, solving large-scale sparse matrices using SciPy’s SuperLU, and defining custom Autograd gradients to bridge physics simulations with automatic differentiation.
]]>With the uncontrolled proliferation of journals, it has become impossible for researchers to keep up with the considerable volume of articles published each month (My doctoral students can confirm this).
Worse still, we are trapped in a system governed by impact factors, where large publishers benefit from free peer review and exorbitant publishing fees. In the age of large language models (LLMs)—where standard text can be generated at virtually no marginal cost—the traditional article is losing its value.
How do we fix a broken model? Here is a roadmap to renew scientific sharing:
💡 1. Reimagining Article Types We need to move beyond rigid and overly text-heavy formats. Mainstream media outlets now value and promote alternative contributions. Data & Software Articles: Treating code and open data sets with equal prestige. Re-Science Articles: Normalizing and rewarding replication studies.
👉 See the model used by the Journal of Open Aviation Studies:JOAS
👉 Distill is going on hiatus “We think that Distill is a really beautiful artifact which illustrates a vision of scientific publishing. But it is not sustainable for us to continue running the journal in its current form. We think preserving it in its present state is more valuable than diluting it with lower quality editing. We also think that it’s a lot healthier for us and frees up our energy to do new projects that provide value to the community… Although Distill is going on hiatus, the Distill template is open source, and we’d love to see others run with it!”
👉 Don’t we have enough journals already? The Journal of Open Source Software JOSS is a developer friendly, open access journal for research software packages. “We built this journal because we believe that after you’ve done the hard work of writing great software, it shouldn’t take weeks and months to write a paper about your work.”
🎵 2. The Music Industry Shift Think about the media evolution: Vinyl ➡️ Napster ➡️ Spotify. Academia went from Paper ➡️ Sci-Hub. Now, we need the final leap: Unified Scientific Streaming Platforms that sustainably support Diamond Open Access (free to read, free to publish) through institutional funding rather than predatory APCs.
👉 See ACM’s forward-looking framework for modern AI science: ACM
🎥 3. Multi-Modal Communication To spark the interest of the next generation of researchers, it is essential to complement dense academic texts with open-source code notebooks and concise video summaries. Even traditional publishing houses are recognizing this shift in communication formats.
👉 Read the recent Nature analysis on this evolution: Nature
The traditional model of scientific journals is an outdated one. It’s time to ditch the metrics and return to the collective advancement of human knowledge for a more sustainable world.
One missing point: AI also helps to find, in the state of the art, unsolved Erdős problems that have not been declared as such in the publication.
👇 How can we renew the way we publish our scientific work?
#OpenScience #AcademicPublishing #GenerativeAI #Research #OpenData #Innovation
]]>Future posts will cover new papers, conference posters, student milestones (PhD defences, new arrivals), and new: a nice way of exploring research through BLOGs and vibe coding EXAMPLES (made with Claude)
]]>