---
kind: concept
slug: adaptive-collocation-sampling
title: Adaptive collocation sampling
summary: >-
  Chooses where a physics-informed network's residual points go by keeping random candidates with the largest PDE error, added to a coarse regular grid. One card covers it so far: in a linear-elasticity study the adaptive variant did not converge from 64^2 points up and did not beat regular sampling overall. Only one variant is sourced, so no alternatives can be compared.
tags: [pinn, sampling, collocation, linear-elasticity]
papers: ["2110.07374"]
related: [physics-informed-neural-network, gradient-clipping]
status: draft
updated_at: 2026-11-08
---

## Overview

A physics-informed neural network (PINN) learns a field by driving the PDE
residual towards zero at a set of collocation points. Regular grids are the
default. Adaptive collocation sampling replaces part of the grid with points
where the current error is largest. The invariant is that the training loss is
still built from residuals at the chosen points, so only the point selection
changes [arxiv:2110.07374 §5.1]. The selection error uses the local PDE
residual only; the global work-balance term is excluded because it cannot be
evaluated per point [arxiv:2110.07374 §5.1].

## Variants

### Residual-driven refinement on a coarse regular grid

Each cycle keeps a fixed coarse regular grid, draws uniform random candidates
over the domain, scores them by local residual, and keeps the worst ones. The
two sets are merged, with the ratio γ = n_reg / n_ada setting how many points
come from each source [arxiv:2110.07374 §5.1]. The paper reports γ = 2.2
[arxiv:2110.07374 §5.1]. Gradient clipping with a threshold α is applied during
these cycles; the paper does not state α [arxiv:2110.07374 §5.1]. The study
uses one regular pre-training pass and one adaptive cycle, each with 5 x 10^3
BFGS iterations [arxiv:2110.07374 §5.3]. Card: [Physics informed neural networks for continuum micromechanics](/papers/2110.07374).

The paper applies this scheme to both a single PINN and a conservative PINN
(cPINN) with a domain split [arxiv:2110.07374 §5.3]. Wight and Zhao (2020) also
propose an adaptive scheme; the paper calls it related but different
[arxiv:2110.07374 §5.1]. That work has no card here, so it is named only.

## When to use which

The only evidence here is one linear-elasticity problem with a single stiff
inclusion, with author-reported results. Treat the statements below as what that study shows, not as general rules.

| Situation | Use | Why |
|---|---|---|
| Dense regular grid is affordable (64^2 or more points) | Regular sampling | The adaptive PINN's mean residual was higher than the regular PINN's at 64^2 and 128^2 [arxiv:2110.07374 §5.3] |
| Very few points (8^2 to 32^2) and a plain PINN | Adaptive sampling may help, check on your problem | Adaptive PINN gave lower mean residual at 16^2 and 32^2, but higher at 8^2 [arxiv:2110.07374 §5.3] |
| Sharp material interface, so one network struggles | Domain decomposition (cPINN) first, then regular sampling | Plain cPINN had lower mean residual than adaptive cPINN at 16^2 and above [arxiv:2110.07374 §5.3] |
| Little time for tuning | Regular sampling | The scheme is sensitive to γ and the number of cycles [arxiv:2110.07374 §5.3] |

## Implementation sketch

Common core of the scheme, with the points that the paper leaves open marked.

```pseudocode
# Adaptive collocation cycles  [arxiv:2110.07374 §5.1]
def adaptive_cycles(net, grid_coarse, n_ada, n_cand, cycles, clip):
    # Pre-train on a fine regular grid first  [arxiv:2110.07374 §5.3]
    train(net, regular_grid_fine())
    for cycle in range(cycles):
        # Candidate count is not stated in the paper  [arxiv:2110.07374 §5.1]
        cand = uniform_points(n_cand)
        # Local PDE residual only; no global work term  [arxiv:2110.07374 §5.1]
        err = local_residual(net, cand)
        # Keep the n_ada points with the largest error
        extra = cand[argsort(-err)[:n_ada]]
        pts = concat(grid_coarse, extra)
        # Gradient clipping threshold alpha is not stated  [arxiv:2110.07374 §5.1]
        train(net, pts, clip_norm=clip)
    return net
```

## Contradictions

Within the one paper, the text and the plotted data disagree. The text says the
adaptive PINN has lower error than the PINN in the first half of the study. The
table shows this only at 16^2 and 32^2; at 8^2 the adaptive PINN is higher
[arxiv:2110.07374 §5.3]. The paper also states that the adaptive variants do not
converge from 64^2 points upwards [arxiv:2110.07374 §5.3]. Other figures are
author-reported and come from one problem, so none can be checked against
another paper here.

## Papers

- [Physics informed neural networks for continuum micromechanics](/papers/2110.07374): adaptive residual-driven sampling compared with regular grids, and a cPINN domain split, on a stiff-inclusion elasticity problem.